Jim Michael Cushing

Interests

Research

Dynamical systems, mathematical biology

Courses

Scholarly Contributions

Books

• Cushing, J. M., Saleem, M., Srivastava, H. M., Kahn, M. A., & Merajuddin, M. (2016). Applied Analysis in Biological and Physical Sciences. Springer Proceedings in Mathematics & Statistics, Volume 186: Springer India.
• Cushing, J. M., Soler, L. A., Elaydi, S., & Pinto, A. A. (2016). Difference Equations, Discrete Dynamical Systems and Applications. Springer Proceedings in Mathematics & Statistics, Volume 180: Springer, Berlin.
• AlSharawi, Z., Cushing, J., & Elaydi, S. (2014). Theory and Applications of Difference Equations and Discrete Dynamical Systems. Springer.
• Elaydi, S., Cushing, J. M., Lasser, R., Ruffing, A., Papageorgiou, V., & van Assche, W. (2007). Difference Equations, Special Funcations and Orthogonal Polynomials. World Scientific Publishing Do. Pte. Ltd. Singapore.
• Cushing, J. M. (2004). Differential Equations: An Applied Approach. Pearson-Prentice Hall.
• Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A., & Henson, S. M. (2003). Chaos in Ecology: Experimental Nonlinear Dynamics. Academic Press, Elsevier Science, New York.
• Cushing, J. M. (1998). An Introduction to Structured Population Dynamics. Conference Series in Applied Mathematics Vol. 71: Society of Industrial and Applied Mathematics.
• Cushing, J. M. (1977). Integrodifferential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomathematics 20: Society of Industrial and Applied Mathematics.

Chapters

• Cushing, J. M. (2019). Stability and the Geometry of Bifurcation Graphs for a Class of Nonlinear Leslie Models. In Difference Equations and Discrete Dynamical Systems with Applications. Springer. doi:DOI: 10.1007/s00285-018-1211-4
• Cushing, J. M. (2016). One dimensional maps as population and evolutionary dynamic models. In Applied Analysis in Biological and Physical Sciences. Springer.
• Cushing, J. M. (2015). On the fundamental bifurcation theorem for semelparous Leslie models. In Mathematics of Planet Earth: Dynamics, Games and Science(pp 199-236). Springer.
• Cushing, J. M. (2014). An evolutionary Beverton-Holt Model. In Theory and Applications of Difference Equations and Discrete Dynamical Systems(pp 127-141). Spring.

Journals/Publications

• Cushing, J. M. (2019). Differene Equations as Models of Evolutionary Population Dynamics. Journal of Biological Dynamics, 16(4), 103-127. doi:10.1080/17513758.2019.1574034
• Cushing, J. M. (2019). Discrete Time Darwinian Dynamics and Semelparity versus Iteroparity. Mathematical Biosciences and Engineering, 16(4), 1815-1835. doi:DOI 10.3934/mbe.2019088
• Cushing, J. M., & Henson, S. M. (2018). Periodic matrix models for seasonal dynamics of structured populations with application to a seabird population. Journal of Mathematical Biology, 77, 1689-1720. doi:DOI: 10.1007/s00285-018-1211-4
• Robertson, S., Henson, S., Robertson, T., & Cushing, J. M. (2018). A matter of maturity: to delay or not to delay? Continuous-time compartmental models of structured populations in the literature 2000-2016. Natural Resource Modeling, 31(1), 1-26. doi:10.1111/nrm.12160
• Cushing, J. M., Martins, F., Pinto, A. A., & Veprauskas, A. (2017). A bifurcation theorem for evolutionary matrix models with multiple traits. Journal of Mathematical Biology, 75(2), 491-520. doi:https://doi.org/10.1007/s0028
• De Leenheer, P., Mohapatra, A., Ohms, H. A., Lytle, D. A., & Cushing, J. M. (2017). The puzzle of partial migration: adaptive dynamics and evolutionary game theory perspectives. Journal of Theoretical Biology, 412, 172-185.
• Verprauskas, A., & Cushing, J. M. (2017). A juvenile-adult population model: climate change, cannibalism, reproductive synchrony, and strong Allee effects. Journal of Biological Dynamics. doi:10.1080/17513758.2015.1131853
• Cushing, J. M., & Diekmann, O. (2016). The many guises of R0 (a didactic note). Journal of Theoretical Biology, 404, 295-302. doi:10.1016/j.jtbi.2016.06.017
• Cushing, J. M., Henson, S. M., & Hayward, J. (2016). Modeling animal behavior in a changing environment. Resource Modeling Association Newsletter.
• Meissen, E. P., Salau, K. R., & Cushing, J. M. (2016). A global bifurcation theorem for Darwinian matrix models. Journal of Difference Equations and Applications. doi:10.1080/10236198.2016.1177522
• Veprauskas, A., & Cushing, J. M. (2016). Evolutionary dynamics of a multi-trait semelparous model. Discrete and Continuous Dynamical Systems Series B, 21(2), 655-767.
• Cushing, J. M. (2015). The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences and Engineering, 12(4), 643-660.
• Cushing, J. M., Ackleh, A. S., & Salceanu, P. L. (2015). On the dynamics of evolutionary competition models. Natural Resource Modeling, 28(4), 380-397. doi:DOI: 10.1111/nrm.12074
• Cushing, J. M., Henson, S. M., & Hayward, J. L. (2015). An evolutionary game theoretic model of cannibalism. Natural Resource Modeling, 28(4), 497-552. doi:10.1111/nrm.12079
• Cushing, J. M. (2014). Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations. Journal of Biological Dynamics, 8, 57-73.
• Cushing, J. M., & Stump, S. M. (2013). Darwinian dynamics of a juvenile-adult model. Mathematical Biosciences and Engineering, 10(4), 1017-1044.
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  PMID: 23906201;Abstract: The bifurcation that occurs from the extinction equilibrium in a basic discrete time, nonlinear juvenile-adult model for semelparous populations, as the inherent net reproductive number R0 increases through 1, exhibits a dynamic dichotomy with two alternatives: an equilibrium with overlapping generations and a synchronous 2-cycle with non-overlapping generations. Which of the two alternatives is stable depends on the intensity of competition between juveniles and adults and on the direction of bifurcation. We study this dynamic dichotomy in an evolutionary setting by assuming adult fertility and juvenile survival are functions of a phenotypic trait u subject to Darwinian evolution. Extinction equilibria for the Darwinian model exist only at traits u* that are critical points of R0 (u). We establish the simultaneous bifurcation of positive equilibria and synchronous 2-cycles as the value of R0 (u*) increases through 1 and describe how the stability of these dynamics depend on the direction of bifurcation, the intensity of between-class competition, and the extremal properties of R0 (u) at u*. These results can be equivalently stated in terms of the inherent population growth rate r (u).
• Allen, L. J., & Cushing, J. M. (2012). Preface. Journal of Biological Dynamics, 6(1), 1-.
• Cushing, J. M. (2012). A dynamic dichotomy for a system of hierarchical difference equations. Journal of Difference Equations and Applications, 18(1), 1-26.
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  Abstract: A system of difference equations that arises in population dynamics is studied. Criteria are given for the existence of equilibria lying in the positive cone and for the existence of periodic cycles lying on the boundary of the cone. These equilibria and cycles arise from a bifurcation that occurs as a fundamental parameter R 0 increases through the value 1. Under monotone conditions on the nonlinearities and for R 0 near 1, we derive criteria for the stability of the equilibria and we determine the global dynamics on the boundary of the cone. We show that boundary orbits tend to periodic cycles (all of which we classify into four types). A dynamic dichotomy is established between the equilibria and the cycles, which asserts that one is stable and the other is unstable. We also establish a dynamic dichotomy between the equilibria and the boundary of the cone. © 2012 Copyright Taylor and Francis Group, LLC.
• Cushing, J. M. (2012). Preface. Journal of Biological Dynamics, 6(SUPPL.2), 1-2.
• Cushing, J. M., & Ackleh, A. S. (2012). A net reproductive number for periodic matrix models. Journal of Biological Dynamics, 6(2), 166-188.
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  PMID: 22873586;Abstract: We give a definition of a net reproductive number R0 for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing-Zhou definition of R0 in the autonomous case. The value of R0 determines whether the population goes extinct (R01). We discuss the biological interpretation of this definition and derive formulas for R0 for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R0 with another definition given recently by Bacaër. © 2012 Copyright Taylor and Francis Group, LLC.
• Cushing, J. M., & Henson, S. M. (2012). Stable bifurcations in semelparous Leslie models. Journal of Biological Dynamics, 6(SUPPL.2), 80-102.
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  PMID: 22937804;Abstract: In this paper, we consider nonlinear Leslie models for the dynamics of semelparous age-structured populations. We establish stability and instability criteria for positive equilibria that bifurcate from the extinction equilibrium at R0=1. When the bifurcation is to the right (forward or super-critical), the criteria consist of inequalities involving the (low-density) between-class and within-class competition intensities. Roughly speaking, stability (respectively, instability) occurs if between-class competition is weaker (respectively, stronger) than within-class competition. When the bifurcation is to the left (backward or sub-critical), the bifurcating equilibria are unstable. We also give criteria that determine whether the boundary of the positive cone is an attractor or a repeller. These general criteria contribute to the study of dynamic dichotomies, known to occur in lower dimensional semelparous Leslie models, between equilibration and age-cohort-synchronized oscillations. © 2012 Copyright J.M. Cushing.
• Cushing, J. M., & Hudson, J. T. (2012). Evolutionary dynamics and strong Allee effects. Journal of Biological Dynamics, 6(2), 941-958.
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  PMID: 22881366;Abstract: We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS. © 2012 Copyright Taylor and Francis Group, LLC.
• J, L., & Cushing, J. M. (2012). Preface. Journal of Biological Dynamics.. Journal of biological dynamics, 6(1), 1-.
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  PMID: 22873519;
• Robertson, S. L., & Cushing, J. M. (2012). A bifurcation analysis of stage-structured density dependent integrodifference equations. Journal of Mathematical Analysis and Applications, 388(1), 490-499.
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  Abstract: There is evidence for density dependent dispersal in many stage-structured species, including flour beetles of the genus Tribolium. We develop a bifurcation theory approach to the existence and stability of (non-extinction) equilibria for a general class of structured integrodifference equation models on finite spatial domains with density dependent kernels, allowing for non-dispersing stages as well as partial dispersal. We show that a continuum of such equilibria bifurcates from the extinction equilibrium when it loses stability as the net reproductive number n increases through 1. Furthermore, the stability of the non-extinction equilibria is determined by the direction of the bifurcation. We provide an example to illustrate the theory. © 2011 Elsevier Inc.
• Robertson, S. L., Cushing, J. M., & Costantino, R. F. (2012). Life Stages: Interactions and Spatial Patterns. Bulletin of Mathematical Biology, 74(2), 491-508.
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  PMID: 22135094;Abstract: In many stage-structured species, different life stages often occupy separate spatial niches in a heterogeneous environment. Life stages of the giant flour beetle Tribolium brevicornis (Leconte), in particular adults and pupae, occupy different locations in a homogeneous habitat. This unique spatial pattern does not occur in the well-studied stored grain pests T. castaneum (Herbst) and T. confusum (Duval). We propose density dependent dispersal as a causal mechanism for this spatial pattern. We model and explore the spatial dynamics of T. brevicornis with a set of four density dependent integrodifference and difference equations. The spatial model exhibits multiple attractors: a spatially uniform attractor and a patchy attractor with pupae and adults spatially separated. The model attractors are consistent with experimental observations. © 2011 Society for Mathematical Biology.
• Cohen, Y., & Cushing, J. M. (2011). Preface. Journal of Biological Dynamics, 5(3), 213-214.
• Cushing, J. M. (2011). On the relationship between r and R₀ and its role in the bifurcation of stable equilibria of Darwinian matrix models. Journal of Biological Dynamics, 5(3), 277-297.
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  Abstract: If the demographic parameters in a matrix model for the dynamics of a structured population are dependent on a parameter u, then the population growth rate r = r(u) and the net reproductive number R0 = R0(u) are functions of u. For a general matrix model, we showthat r and R0 share critical values and extrema at values u = u* for which r(u*) = R0(u*) = 1. This allows us to re-interpret, in terms of the more analytically tractable quantity R0, a fundamental bifurcation theorem for non-linear Darwinian matrix models from the evolutionary game theory that concerns the destabilization of the extinction equilibrium and creation of positive equilibria. Two illustrations are given: a theoretical study of trade-offs between fertility and survivorship in the evolution of an evolutionarily stable strategies and an application to an experimental study of the evolution to a genetic polymorphism. © 2011 Taylor & Francis.
• Henson, S. M., Cushing, J. M., & Hayward, J. L. (2011). Socially induced ovulation synchrony and its effect on seabird population dynamics. Journal of Biological Dynamics, 5(5), 495-516.
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  Abstract: Spontaneous oscillator synchrony is a form of self-organization in which populations of interacting oscillators ultimately cycle together. This phenomenon occurs in a wide range of physical and biological systems. In rats and humans, oestrous/menstrual cycles synchronize through social stimulation with pheromones acting as synchronizing signals. In previous work, we showed that glaucous-winged gulls (Larus glaucescens) can lay eggs synchronously on an every-other-day schedule, and that synchrony increases with colony density. We posed a discrete-time mathematical model for reproduction during the breeding season based on the hypothesis that pre-ovulatory luteinizing hormone surges synchronize by means of visual, auditory and/or olfactory cues. Here, we extend the seasonal model in order to investigate the effect of ovulation synchrony on population dynamics across reproductive seasons. We show that socially stimulated ovulation synchrony can enhance total population size and allow the population to persist at lower birth rates than would otherwise be possible. © 2011 Taylor & Francis.
• Jia, L. i., Cushing, J., & Elaydi, S. (2011). Preface. Journal of Biological Dynamics, 5(5), 367-.
• Rael, R. C., Vincent, T. L., & Cushing, J. M. (2011). Competitive outcomes changed by evolution. Journal of Biological Dynamics, 5(3), 227-252.
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  Abstract: When evolution plays a role, population dynamic models alone are not sufficient for determining the outcome of multi-species interactions.As an expansion of Maynard Smith's concept of an evolutionarily stable strategy, evolutionary game theory (EGT) combines population and evolutionary dynamics so that natural selection becomes a dynamic game with phenotypic traits as strategies. Using EGT,we showhowevolution can change the outcome of the interaction between two competing species.We are particularly interested in situations where the interaction changes from one of competitive exclusion to one of coexistence and where the interaction changes from the exclusion of one species to the exclusion of the other. There are two crucial factors that determine whether or not such events can occur: the rate of evolution (which is proportional to the amount of variability in phenotypic traits within a species) and what we term the boxer effect, a phenomenon that concerns how the intensity of interspecies competition relates to species' trait differences. We apply the theory to data from two historical competition experiments involving species of flour beetle from the genus Tribolium. In both applications, we show how EGT can explain certain seemingly 'anomalous' experimental results. © 2011 Taylor & Francis.
• Robertson, S. L., & Cushing, J. M. (2011). Spatial segregation in stage-structured populations with an application to Tribolium. Journal of Biological Dynamics, 5(5), 398-409.
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  Abstract: Spatial segregation among life-cycle stages has been observed in many stage-structured species, including species of the flour beetle Tribolium. We investigate density-dependent dispersal of life-cycle stages as a possible mechanism responsible for this separation. We explore this hypothesis using stage-structured, integrodifference equation (IDE) models that incorporate density-dependent dispersal kernels. We first investigate mechanisms that can lead to spatial patterns in juvenile-adult IDE models. We show, via numerical simulation, that density-dependent dispersal can lead to the spatial segregation of life-cycle stages in the sense that each stage peaks in a different spatial location. We then construct a three-stage spatial model to describe the population dynamics of Tribolium castaneum and Tribolium confusum and assess density-dependent dispersal mechanisms that are able to explain spatial patterns that have been experimentally observed in these species. © 2011 Taylor & Francis.
• Cushing, J. M. (2010). On the dynamics of a class of Darwinian matrix models. Nonlinear Dynamics and Systems Theory, 10(2), 103-116.
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  Abstract: Using the methodology of evolutionary game theory (EGT), I study a class of Darwinian matrix models which are derived from a class of nonlinear matrix models for structured populations that are known to possess stable (normalized) distributions. Utilizing the limiting equations that result from this ergodic property, I prove extinction and stability results for the limiting equations of the EGT versions of these kinds of structured population models. This is done in a bifurcation theory context. The results provide conditions sufficient for a branch of non-extinction equilibria to bifurcate from the branch of extinction equilibria. When this bifurcation is supercritical (explicit criteria are given), these equilibria are stable and represent stable non-extinction equilibria (which are also candidate ESS equilibria). These kinds of matrix models are motivated by applications to size structured populations, and I give an application of this type. Besides illustrating the formal theory, this application shows the importance of trade-offs among life history parameters that are necessary for the existence of an evolutionarily stable equilibrium. © 2010 InforMath Publishing Group.
• Henson, S. M., Hayward, J. L., Cushing, J. M., & Galusha, J. C. (2010). Socially induced synchronization of every-other-day egg laying in a seabird colony. Auk, 127(3), 571-580.
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  Abstract: Spontaneous oscillator synchrony has been documented in a wide variety of electrical, mechanical, chemical, and biological systems, including the menstrual cycles of women and estrous cycles of Norway Rats (Rattus norvegicus). In temperate regions, many colonial birds breed seasonally in a time window set by photoperiod; some studies have suggested that heightened social stimulation in denser colonies can lead to a tightened annual reproductive pulse. It has been unknown, however, whether the analog of menstrual synchrony occurs in birdsthat is, whether avian ovulation cycles can synchronize on a daily timescale within the annual breeding pulse. We report every-other-day clutch-initiation and egg-laying synchrony in a breeding colony of Glaucous-winged Gulls (Larus glaucescens) and show that the level of synchrony declined with decreasing colony density. We also pose a mathematical model based on the hypothesis that preovulatory luteinizing hormone surges synchronize through social stimulation. Model predictions are consistent with observations. Finally, we suggest a procedure for identifying synchronous egg laying in other colonies and species. © 2010 by The American Ornithologists' Union. All rights reserved.
• Cushing, J. M. (2009). Three stage semelparous Leslie models. Journal of Mathematical Biology, 59(1), 75-104.
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  PMID: 18777023;Abstract: Nonlinear Leslie matrix models have a long history of use for modeling the dynamics of semelparous species. Semelparous models, as do nonlinear matrix models in general, undergo a transcritical equilibrium bifurcation at inherent net reproductive number R 0 = 1 where the extinction equilibrium loses stability. Semelparous models however do not fall under the purview of the general theory because this bifurcation is of higher co-dimension. This mathematical fact has biological implications that relate to a dichotomy of dynamic possibilities, namely, an equilibration with over lapping age classes as opposed to an oscillation in which age classes are periodically missing. The latter possibility makes these models of particular interest, for example, in application to the well known outbreaks of periodical insects. While the nature of the bifurcation at R 0 = 1 is known for two-dimensional semelparous Leslie models, only limited results are available for higher dimensional models. In this paper I give a thorough accounting of the bifurcation at R 0 = 1 in the three-dimensional case, under some monotonicity assumptions on the nonlinearities. In addition to the bifurcation of positive equilibria, there occurs a bifurcation of invariant loops that lie on the boundary of the positive cone. I describe the geometry of these loops, classify them into three distinct types, and show that they consist of either one or two three-cycles and heteroclinic orbits connecting (the phases of) these cycles. Furthermore, I determine stability and instability properties of these loops, in terms of model parameters, as well as those of the positive equilibria. The analysis also provides the global dynamics on the boundary of the cone. The stability and instability conditions are expressed in terms of certain measures of the strength and the symmetry/asymmetry of the inter-age class competitive interactions. Roughly speaking, strong inter-age class competitive interactions promote oscillations (not necessarily periodic) with separated life-cycle stages, while weak interactions promote stable equilibration with overlapping life-cycle stages. Methods used include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques. © 2008 Springer-Verlag.
• Cushing, J., & Cushing, J. M. (2009). Three stage semelparous Leslie models. Journal of mathematical biology, 59(1).
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  Nonlinear Leslie matrix models have a long history of use for modeling the dynamics of semelparous species. Semelparous models, as do nonlinear matrix models in general, undergo a transcritical equilibrium bifurcation at inherent net reproductive number R(0) = 1 where the extinction equilibrium loses stability. Semelparous models however do not fall under the purview of the general theory because this bifurcation is of higher co-dimension. This mathematical fact has biological implications that relate to a dichotomy of dynamic possibilities, namely, an equilibration with over lapping age classes as opposed to an oscillation in which age classes are periodically missing. The latter possibility makes these models of particular interest, for example, in application to the well known outbreaks of periodical insects. While the nature of the bifurcation at R(0) = 1 is known for two-dimensional semelparous Leslie models, only limited results are available for higher dimensional models. In this paper I give a thorough accounting of the bifurcation at R(0) = 1 in the three-dimensional case, under some monotonicity assumptions on the nonlinearities. In addition to the bifurcation of positive equilibria, there occurs a bifurcation of invariant loops that lie on the boundary of the positive cone. I describe the geometry of these loops, classify them into three distinct types, and show that they consist of either one or two three-cycles and heteroclinic orbits connecting (the phases of) these cycles. Furthermore, I determine stability and instability properties of these loops, in terms of model parameters, as well as those of the positive equilibria. The analysis also provides the global dynamics on the boundary of the cone. The stability and instability conditions are expressed in terms of certain measures of the strength and the symmetry/asymmetry of the inter-age class competitive interactions. Roughly speaking, strong inter-age class competitive interactions promote oscillations (not necessarily periodic) with separated life-cycle stages, while weak interactions promote stable equilibration with overlapping life-cycle stages. Methods used include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques.
• Rael, R. C., Costantino, R. F., Cushing, J. M., & Vincent, T. L. (2009). Using stage-structured evolutionary game theory to model the experimentally observed evolution of a genetic polymorphism. Evolutionary Ecology Research, 11(2), 141-151.
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  Abstract: Aim: Our aim is to show the utility of evolutionary game theory (EGT) methods in describing and predicting the outcome of experiments for which genetic data are available in the absence of phenotypic data. As an example we use experimental data from genetically perturbed cultures of the flour beetle Tribolium castaneum. Using natural selection, the theory provides a theoretical basis for the observed population dynamics and evolution of a polymorphism of wild-type and corn oil sensitive alleles. Method of analysis: We derive an EGT version of a well-validated model for the population dynamics of Tribolium (the LPA model) using a version of EGT developed for stage-structured populations given by Vincent and Brown (2005). We use the wild-type allele frequency as the strategy in this model. We estimate model parameters and conduct simulations using the parameterized model. Experimental data: We use population and genetic data from an experiment conducted by Desharnais (1979). Cultures of T. castaneum homozygous for corn oil sensitivity are perturbed by adding homozygous wild-type individuals. The data include population densities of larvae, pupae, and adults as well as allele frequencies obtained from genetically perturbed cultures. Conclusions: The parameterized EGT version of the LPA model fits reasonably accurately the population and genetic data in both the control and genetically perturbed treatments. For both treatments, the model predicts an oscillatory 2-cycle asymptotic attractor. The predicted oscillations match well with the oscillations observed in the population data, even capturing observed increases in amplitude and population densities in the genetically perturbed treatment. Interestingly, the model also predicts a periodic oscillation in allele frequency, although the amplitude is too small to be detected in the data. We conclude that the EGTpredicted evolution of a polymorphism by natural selection is obtained in agreement with the experimental data. Our results illustrate two points concerning EGT: first, the theory can be successful in application to real data and, second, it can be applied in a context that allows a connection to classical population genetics theory. © 2009 R.C. Rael.
• Arpin, S. L., & Cushing, J. M. (2008). Modeling frequency-dependent selection with an application to cichlid fish. Mathematical Biosciences and Engineering, 5(4), 889-903.
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  PMID: 19278289;Abstract: Negative frequency-dependent selection is a well known microevolutionary process that has been documented in a population of Perissodus microlepis, a species of cichlid fish endemic to Lake Tanganyika (Africa). Adult P. microlepis are lepidophages, feeding on the scales of other living fish. As an adaptation for this feeding behavior P. microlepis exhibit lateral asymmetry with respect to jaw morphology: the mouth either opens to the right or left side of the body. Field data illustrate a temporal phenotypic oscillation in the mouth-handedness, and this oscillation is maintained by frequency-dependent selection. Since both genetic and population dynamics occur on the same time scale in this case, we develop a (discrete time) model for P. microlepis populations that accounts for both dynamic processes. We establish conditions on model parameters under which the model predicts extinction and conditions under which there exists a unique positive (survival) equilibrium. We show that at the positive equilibrium there is a 1:1 phenotypic ratio. Using a Local stability and bifurcation analysis, we give further conditions under which the positive equilibrium is stable and conditions under which it is unstable. Destabilization results in a bifurcation to a periodic oscillation and occurs when frequency-dependent selection is sufficiently strong. This bifurcation is offered as an explanation of the phenotypic frequency oscillations observed in P. microlepis. An analysis of the bifurcating periodic cycle results in some interesting and unexpected predictions.
• Chitnis, N., Hyman, J. M., & Cushing, J. M. (2008). Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology, 70(5), 1272-1296.
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  PMID: 18293044;Abstract: We perform sensitivity analyses on a mathematical model of malaria transmission to determine the relative importance of model parameters to disease transmission and prevalence. We compile two sets of baseline parameter values: one for areas of high transmission and one for low transmission. We compute sensitivity indices of the reproductive number (which measures initial disease transmission) and the endemic equilibrium point (which measures disease prevalence) to the parameters at the baseline values. We find that in areas of low transmission, the reproductive number and the equilibrium proportion of infectious humans are most sensitive to the mosquito biting rate. In areas of high transmission, the reproductive number is again most sensitive to the mosquito biting rate, but the equilibrium proportion of infectious humans is most sensitive to the human recovery rate. This suggests strategies that target the mosquito biting rate (such as the use of insecticide-treated bed nets and indoor residual spraying) and those that target the human recovery rate (such as the prompt diagnosis and treatment of infectious individuals) can be successful in controlling malaria. © 2008 Society for Mathematical Biology.
• Cushing, J., Cushing, J. M., & Arpin, S. L. (2008). Modeling frequency-dependent selection with an application to cichlid fish. Mathematical biosciences and engineering : MBE, 5(4).
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  Negative frequency-dependent selection is a well known microevolutionary process that has been documented in a population of Perissodus microlepis, a species of cichlid fish endemic to Lake Tanganyika (Africa). Adult P. microlepis are lepidophages, feeding on the scales of other living fish. As an adaptation for this feeding behavior P. microlepis exhibit lateral asymmetry with respect to jaw morphology: the mouth either opens to the right or left side of the body. Field data illustrate a temporal phenotypic oscillation in the mouth-handedness, and this oscillation is maintained by frequency-dependent selection. Since both genetic and population dynamics occur on the same time scale in this case, we develop a (discrete time) model for P. microlepis populations that accounts for both dynamic processes. We establish conditions on model parameters under which the model predicts extinction and conditions under which there exists a unique positive (survival) equilibrium. We show that at the positive equilibrium there is a 1:1 phenotypic ratio. Using a local stability and bifurcation analysis, we give further conditions under which the positive equilibrium is stable and conditions under which it is unstable. Destabilization results in a bifurcation to a periodic oscillation and occurs when frequency-dependent selection is sufficiently strong. This bifurcation is offered as an explanation of the phenotypic frequency oscillations observed in P. microlepis. An analysis of the bifurcating periodic cycle results in some interesting and unexpected predictions.
• Cushing, J. M., Henson, S. M., & Blackburn, C. C. (2007). Multiple mixed-type attractors in a competition model.. Journal of biological dynamics, 1(4), 347-362.
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  PMID: 22876821;Abstract: We show that a discrete-time, two-species competition model with Ricker (exponential) nonlinearities can exhibit multiple mixed-type attractors. By this is meant dynamic scenarios in which there are simultaneously present both coexistence attractors (in which both species are present) and exclusion attractors (in which one species is absent). Recent studies have investigated the inclusion of life-cycle stages in competition models as a casual mechanism for the existence of these kinds of multiple attractors. In this paper we investigate the role of nonlinearities in competition models without life-cycle stages.
• Cushing, J. M., Henson, S. M., & Roeger, L. (2007). Coexistence of competing juvenile-adult structured populations.. Journal of biological dynamics, 1(2), 201-231.
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  PMID: 22873341;Abstract: The Leslie-Gower model is a discrete time analog of the competition Lotka-Volterra model and is known to possess the same dynamic scenarios of that famous model. The Leslie-Gower model played a historically significant role in the history of competition theory in its application to classic laboratory experiments of two competing species of flour beetles (carried out by Park in the 1940s-1960s). While these experiments generally supported what became the Competitive Exclusion Principle, Park observed an anomalous coexistence case. Recent literature has discussed Park's 'coexistence case' by means of non-Lotka-Volterra, non-equilibrium dynamics that occur in a high dimensional model with life cycle stages. We study this dynamic possibility in the lowest possible dimension, that is to say, by means of a model involving only two species each with two life cycle stages. We do this by extending the Leslie-Gower model so as to describe the competitive interaction of two species with juvenile and adult classes. We give a complete account of the global dynamics of the resulting model and show that it allows for non-equilibrium competitive coexistence as competition coefficients are increased. We also show that this phenomenon occurs in a general class of models for competing populations structured by juvenile and adult life cycle stages.
• Henson, S. M., Dennis, B., Hayward, J. L., Cushing, J. M., & Galusha, J. G. (2007). Predicting the dynamics of animal behaviour in field populations. Animal Behaviour, 74(1), 103-110.
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  Abstract: Many species show considerable variation in behaviour among individuals. We show that some behaviours are largely deterministic and predictable with mathematical models. We propose a general differential equation model of behaviour in field populations and use the methodology to explain and predict the dynamics of sleep and colony attendance in seabirds as a function of environmental factors. Our model explained over half the variability in the data to which it was fitted, and it predicted the dynamics of an independent data set. Differential equation models may provide new approaches to the study of behaviour in animals and humans. © 2007 The Association for the Study of Animal Behaviour.
• Henson, S. M., Galusha, J. G., Hayward, J. L., & Cushing, J. M. (2007). Modeling territory attendance and preening behavior in a seabird colony as functions of environmental conditions.. Journal of biological dynamics, 1(1), 95-107.
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  PMID: 22880615;Abstract: In previous studies we developed a general compartmental methodology for modeling animal behavior and applied the methodology to marine birds and mammals. In this study we used the methodology to construct a system of two differential equations to model the dynamics of territory attendance and preening in a gull colony on Protection Island, Strait of Juan de Fuca, Washington. We found that colony occupancy was driven primarily by abiotic environmental conditions, including tide height, time of day, solar elevation, and wind speed over open water. For birds in the colony, preening behavior was driven to some extent by abiotic environmental conditions (including time of day, solar elevation, humidity, and wind speed on the colony), but apparently was driven primarily by local and/or biotic effects not included in the model. In terms of R(2) values, the model explained 65% and 37% of the variability in colony occupancy and preening data, respectively, as a function of these six abiotic environmental factors.
• Chitnis, N., Cushing, J. M., & Hyman, J. M. (2006). Bifurcation analysis of a mathematical model for malaria transmission. SIAM Journal on Applied Mathematics, 67(1), 24-45.
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  Abstract: We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, and recovered classes, before reentering the susceptible class. Susceptible mosquitoes can become infected when they bite infectious or recovered humans, and once infected they move through the exposed and infectious classes. Both species follow a logistic population model, with humans having immigration and disease-induced death. We define a reproductive number, R0, for the number of secondary cases that one infected individual will cause through the duration of the infectious period. We find that the disease-free equilibrium is locally asymptotically stable when R 0 < 1 and unstable when R0 > 1. We prove the existence of at least one endemic equilibrium point for all R0 > 1. In the absence of disease-induced death, we prove that the transcritical bifurcation at R0 = 1 is supercritical (forward). Numerical simulations show that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at R0 = 1. © 2006 Society for Industrial and Applied Mathematics.
• Cushing, J. M. (2006). A juvenile-adult model with periodic vital rates. Journal of Mathematical Biology, 53(4), 520-539.
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  PMID: 16636871;Abstract: A global branch of positive cycles is shown to exist for a general discrete time, juvenile-adult model with periodically varying coefficients. The branch bifurcates from the extinction state at a critical value of the mean, inherent fertility rate. In comparison to the autonomous system with the same mean fertility rate, the critical bifurcation value can either increase or decrease with the introduction of periodicities. Thus, periodic oscillations in vital parameter can be either advantageous or deleterious. A determining factor is the phase relationship among the oscillations in the inherent fertility and survival rates.
• Cushing, J. M. (2006). Nonlinear semelparous Leslie models. Mathematical Biosciences and Engineering, 3(1), 17-36.
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  PMID: 20361805;Abstract: In this paper we consider the bifurcations that occur at the trivial equilibrium of a general class of nonlinear Leslie matrix models for the dynamics of a structured population in which only the oldest class is reproductive. Using the inherent net reproductive number n as a parameter, we show that a global branch of positive equilibria bifurcates from the trivial equilibrium at n = 1 despite the fact that the bifurcation is nongeneric. The bifurcation can be either supercritical or subcritical, but unlike the case of a generic transcritical bifurcation in iteroparous models, the stability of the bifurcating positive equilibria is not determined by the direction of bifurcation. In addition we show that a branch of single-class cycles also bifurcates from the trivial equilibrium at n = 1. In the case of two population classes, either the bifurcating equilibria or the bifurcating cycles are stable (but not both) depending on the relative strengths of the inter- and intra-class competition. Strong inter-class competition leads to stable cycles in which the two population classes are temporally separated. In the case of three or more classes the bifurcating cycles often lie on a bifurcating invariant loop whose structure is that of a cycle chain consisting of the different phases of a periodic cycle connected by heteroclinic orbits. Under certain circumstances, these bifurcating loops are attractors.
• Cushing, J., & Cushing, J. M. (2006). A juvenile-adult model with periodic vital rates. Journal of mathematical biology, 53(4).
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  A global branch of positive cycles is shown to exist for a general discrete time, juvenile-adult model with periodically varying coefficients. The branch bifurcates from the extinction state at a critical value of the mean, inherent fertility rate. In comparison to the autonomous system with the same mean fertility rate, the critical bifurcation value can either increase or decrease with the introduction of periodicities. Thus, periodic oscillations in vital parameter can be either advantageous or deleterious. A determining factor is the phase relationship among the oscillations in the inherent fertility and survival rates.
• Cushing, J., & Cushing, J. M. (2006). Nonlinear semelparous leslie models. Mathematical biosciences and engineering : MBE, 3(1).
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  In this paper we consider the bifurcations that occur at the trivial equilibrium of a general class of nonlinear Leslie matrix models for the dynamics of a structured population in which only the oldest class is reproductive. Using the inherent net reproductive number n as a parameter, we show that a global branch of positive equilibria bifurcates from the trivial equilibrium at n = 1 despite the fact that the bifurcation is nongeneric. The bifurcation can be either supercritical or subcritical, but unlike the case of a generic transcritical bifurcation in iteroparous models, the stability of the bifurcating positive equilibria is not determined by the direction of bifurcation. In addition we show that a branch of single-class cycles also bifurcates from the trivial equilibrium at n = 1. In the case of two population classes, either the bifurcating equilibria or the bifurcating cycles are stable (but not both) depending on the relative strengths of the inter- and intra-class competition. Strong inter-class competition leads to stable cycles in which the two population classes are temporally separated. In the case of three or more classes the bifurcating cycles often lie on a bifurcating invariant loop whose structure is that of a cycle chain consisting of the different phases of a periodic cycle connected by heteroclinic orbits. Under certain circumstances, these bifurcating loops are attractors.
• Desharnais, R. A., Costantino, R. F., Cushing, J. M., Henson, S. M., Dennis, B., & King, A. A. (2006). Experimental support of the scaling rule for demographic stochasticity. Ecology Letters, 9(5), 537-547.
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  PMID: 16643299;Abstract: A scaling rule of ecological theory, accepted but lacking experimental confirmation, is that the magnitude of fluctuations in population densities due to demographic stochasticity scales inversely with the square root of population numbers. This supposition is based on analyses of models exhibiting exponential growth or stable equilibria. Using two quantitative measures, we extend the scaling rule to situations in which population densities fluctuate due to nonlinear deterministic dynamics. These measures are applied to populations of the flour beetle Tribolium castaneum that display chaotic dynamics in both 20-g and 60-g habitats. Populations cultured in the larger habitat exhibit a clarification of the deterministic dynamics, which follows the inverse square root rule. Lattice effects, a deterministic phenomenon caused by the discrete nature of individuals, can cause deviations from the scaling rule when population numbers are small. The scaling rule is robust to the probability distribution used to model demographic variation among individuals. © 2006 Blackwell Publishing Ltd/CNRS.
• Costantino, R. F., Desharnais, R. A., Cushing, J. M., Dennis, B., Henson, S. M., & King, A. A. (2005). Nonlinear Stochastic Population Dynamics: The Flour Beetle Tribolium as an Effective Tool of Discovery. Advances in Ecological Research, 37, 101-141.
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  Abstract: Prior to the seminal work of R.M. May in the 1970s, the prevailing paradigm viewed the unpredictable fluctuations in population time-series data as random effects due to environmental variability and/or measurement errors. In the absence of environmental variability, according to this view, population numbers would either equilibrate or settle into regular periodic oscillations. May's (1974) suggestion that simple deterministic rules might explain the complex fluctuations observed in animal abundances led to an intense search for chaos in extant population data. The results of the search were suggestive, but equivocal, and May's hypothesis remained the subject of lively debate (Zimmer, 1999; Perry et al., 2000). We took a different approach. Our interdisciplinary research team composed of statisticians, mathematicians, and biologists gathered together in the early 1990s to document experimentally the occurrence of nonlinear dynamic phenomena in biological populations. We began with the idea that nonlinear theory yields testable hypotheses concerning changes in the dynamical behaviors of populations. For example, in the case of the quadratic map (sometimes called the "logistic" map), changes in the intrinsic growth rate lead to a sequence of dynamical behaviors from equilibria to periodic cycles, to aperiodic chaotic behavior. Our thought was that a sequence of changes in dynamical behavior, which is a common feature of nonlinear models, could be tested, in principle, under controlled laboratory conditions. This would provide a connection between theory and data that was missing from ecology. From the very beginning of our collaboration, fundamental questions greeted us at every turn as we looked at historical time-series data and at data collected in our laboratories. Over the years we struggled to combine deterministic concepts such as equilibria, cycles, saddle nodes, bifurcations, basins of attraction, multiple attractors, resonance, and chaos with observations. What would a stable equilibrium, let alone chaos, look like in a population? Could a saddle node be invoked as an explanation for different transient behaviors of time series among replicate populations? Is chaos even possible if we consider discrete-state population models? Is it useful to consider populations as discrete-state stochastic systems? In ecological theory, a central (and abiding) problem is to situate deterministic theory in the context of biological systems where important demographic events are probabilistic. Chance variation, in such fundamental biological processes as the number of offspring per adult and the chance of an individual surviving to adulthood, is a part of population dynamics. Probabilistic variation enters the overall research effort in the statistical methods associated with model identification, parameter estimation, and model validation. Chance events are also a component of the interpretation of population behavior; probabilistic variation is essential to the explanation of ecological time-series data. We expand on these points. First, a mathematical population model, built and tested as a serious scientific hypothesis, must be somehow connected to data. A probabilistic version of the model must be constructed to account for inevitable deviations of data from the predictions of the deterministic model. Demographic/environmental variability must be modeled in order to construct an appropriate estimating function for the model parameters (based on the likelihood or conditional sums of squares, for instance). Statistical diagnostic procedures should be used to evaluate the uncertainty component of the model. Second, chance events interact with deterministic forces to produce emergent dynamic behaviors. The deterministic skeleton fixes the geometry of state space, providing a stage for the transient dance of stochasticity. Chance events allow the system to visit (and re-visit) the various deterministic entities on the stage, including unstable invariant sets, which under strict deterministic theory would have little or no impact on population time series. Ecological time series can display a stochastic mix of many of the dynamic features of the skeleton, including multiple attractors, transients, unstable invariant sets (such as saddles and unstable manifolds) and lattice effects. Stochasticity enlarges the repertoire of time-series orbits; each population, even in a set of laboratory replicates, may display a unique sequence of population abundances. In this chapter, we expand upon the message that in order to understand population fluctuations, deterministic and stochastic forces must be viewed as an integral part of the ecological system. We begin by explaining our models, both animal and mathematical. We then discuss how we estimated model parameters and validated the model. Next, with the parameterized model in hand, we present an overview of some of the nonlinear phenomena and related topics that we have documented in our experimental system: chaotic dynamics, population outbreaks, saddle nodes, phase switching, lattice effects, the anatomy of chaos and, finally, mechanistic models of stochasticity. © 2005 Elsevier Inc. All rights reserved.
• Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A., & Henson, S. M. (2005). Chaos in Ecology. Chaos in Ecology.
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  Abstract: It is impossible to predict the exact behavior of all biological systems and how these same systems are exemplified by patterns of complexity and regularity. Decades of research in ecology have documented how these sorts of patterns are the consequences of deceptively simple rules that determine the nature of the patterns created. Chaos in Ecology will explain how simple beginnings result in complicated results. Chaos in Ecology is the inaugural volume of Theoretical Ecology Series. The authors of this volume have employed data from a proven model system in population dynamics. As a result, this book will be of interest to anyone interested in the ecology of populations. It is impossible to predict the exact behavior of almost all biological systems and yet these same systems are exemplified by patterns of complexity and regularity. Decades of research in ecology have documented that these sorts of patterns are the consequence of deceptively simple rules that determine the nature of the patterns created. In essence, simple beginnings result in complicated results. This realization is captured in the mathematical notion of "chaos" and is rendered intuitive by the oft-repeated metaphor: "A butterfly beats its wings in China and causing a thunderstorm in the Midwest." Thus, seemingly trivial initial conditions (e.g. a butterfly in China) cascade through a series of intermediate events to create a significant large-scale event (e.g. a thunderstorm). Chaos in Ecology is the inaugural volume of Theoretical Ecology Series. The authors of this volume have employed data from a proven model system in population dynamics. As a result, this book will be of interest to anyone interested in the ecology of populations. Chaos in Ecology is a convincing demonstration of chaos in a biological population. The book synthesizes an ecologically focused interdisciplinary blend of non-linear dynamics theory, statistics, and experimentation yielding results of uncommon clarity and rigor. Topics include fundamental issues that are of general and widespread importance to population biology and ecology. Detailed descriptions are included of the mathematical, statistical, and experimental steps they used to explore nonlinear dynamics in ecology. Beginning with a brief overview of chaos theory and its implications for ecology. The book continues by deriving and rigorously testing a mathematical model that is closely wedded to biological mechanisms of their research organism. Therefrom were generated a variety of predictions that are fundamental to chaos theory and experiments were designed and analyzed to test those predictions. Discussion of patterns in chaos and how they can be investigated using real data follows and book ends with a discussion of the salient lessons learned from this research program Book jacket. "The authors have made very circumspect model verifications...The greatest merit of the book is that it confirms experimentally the existence of numerous nonlinear phenomena in populations and presents a new characteristic of populations, that is, the lattice effect. The book is written in an excellent style with high mathematical precision. This work is warmly recommended to all theoretical and field ecologists, and all scientists interested in nonlinear dynamics." -COMMUNITY ECOLOGY (2005) "...rich in statistical concepts and methods. Anyone interested in the dynamics of populations will benefit from reading this book." -ECOLOGY (2003) "In a very real sense, Cushing et al. have done for ecology what Newton did for physics. They have demonstrated that the observed, very complex behavior of biological populations can, at least in principle, be derived and predicted from a knowledge of the operant biological mechanisms expressed in simple mathematical models. ...Chaos in Ecology is a road-map for making ecology into a genuinely predictive science." -Aaron A. King, University of Tennessee (2002) "This is the definitive source on chaos in ecology. Well-written and authoritative." -Simon Levin, Princeton University (2002) "Chaos in Ecology ought to be read by both field and theoretical ecologists." -COMPLEX SYSTEMS LAB, BARCELONA SPAIN. "This work is warmly recommended to all theoretical and field ecologists, and all scientists interested in nonlinear dynamics." -COMMUNITY ECOLOGY (2005) "...an elegant blend of mathematics, statistics, and biology..." -CHANCE (Fall 2003) "...rich in statistical concepts and methods. Anyone interested in the dynamics of populations will benefit from reading this book." -ECOLOGY (2003) "In a very real sense, Cushing et al. have done for ecology what Newton did for physics." -Aaron A. King, University of Tennessee (2002) "This is the definitive source on chaos in ecology. Well-written and authoritative." -Simon Levin, Princeton University (2002) "Chaos in Ecology ought to be read by both field and theoretical ecologists." -COMPLEX SYSTEMS LAB, BARCELONA SPAIN. © 2005 Elsevier Inc. All rights reserved.
• Jang, S. R., & Cushing, J. M. (2005). Dynamics of hierarchical models in discrete-time. Journal of Difference Equations and Applications, 11(2), 95-115.
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  Abstract: We derive and analyze a general class of difference equation models for the dynamics of hierarchically organized populations. Different forms of intra-specific competition give rise to different types of nonlinearities. For our models, we prove that contest competition results asymptotically in only equilibrium dynamics. Scramble competition, on the other hand, can result in more complex asymptotic dynamics. We study both the case when the limiting resource is a constant and when it is dynamically modeled. We prove, in all cases, that the population persists if the inherent net reproductive number of the population is greater than one.
• Cushing, J. M., Levarge, S., Chitnis, N., & Henson, S. M. (2004). Some discrete competition models and the competitive exclusion principle. Journal of Difference Equations and Applications, 10(13-15), 1139-1151.
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  Abstract: A difference equation model, called that Leslie/Gower model, played a key historical role in laboratory experiments that helped establish the "competitive exclusion principle" in ecology. We show that this model has the same dynamic scenarios as the famous Lotka/Volterra (differential equation) competition model. It is less well known that some anomalous results from the experiments seem to contradict the exclusion principle and Lotka/Volterra dynamics. We give an example of a competition model that has non-Lotka/Volterra dynamics that are consistent with the anomalous case.
• King, A. A., Costantino, R. F., Cushing, J. M., Henson, S. M., Desharnais, R. A., & Dennis, B. (2004). Anatomy of a chaotic attractor: Subtle model-predicted patterns revealed in population data. Proceedings of the National Academy of Sciences of the United States of America, 101(1), 408-413.
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  PMID: 14681555;PMCID: PMC314198;Abstract: Mathematically, chaotic dynamics are not devoid of order but display episodes of near-cyclic temporal patterns. This is illustrated, in interesting ways, in the case of chaotic biological populations. Despite the individual nature of organisms and the noisy nature of biological time series, subtle temporal patterns have been detected. By using data drawn from chaotic insect populations, we show quantitatively that chaos manifests itself as a tapestry of identifiable and predictable patterns woven together by stochasticity. We show too that the mixture of patterns an experimentalist can expect to see depends on the scale of the system under study.
• Cushing, J. M. (2003). Cycle chains and the LPA model. Journal of Difference Equations and Applications, 9(7), 655-670.
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  Abstract: The LPA model is a three dimensional system of nonlinear difference equations that has found many applications in population dynamics and ecology. In this paper, we consider a special case of the model (a case that approximates that used in experimental bifurcation and chaos studies) and prove several theorems concerning the existence and stability of periodic cycles, invariant loops, and chaos. Key is the notion of synchronous orbits (i.e. orbits lying on the coordinate planes). The main result concerns the existence of an invariant loop of synchronous orbits that bifurcates, in a nongeneric way, from the trivial equilibrium. The geometry and dynamics of this invariant loop are characterized. Specifically, it is shown that the loop is a cycle chain consisting of synchronous heteroclinic orbits connecting the three temporal phases of a synchronous 3-cycle. We also show that a period doubling route to chaos occurs within the class of synchronous orbits.
• Cushing, J. M., & Henson, S. M. (2003). Erratum: A periodically forced Beverton-Holt equation (Journal of Difference Equations and Applications (2002) 8:12 (1119-1120)). Journal of Difference Equations and Applications, 9(6), 613-.
• Dennis, B., Desharnais, R. A., Cushing, J. M., Henson, S. M., & Costantino, R. F. (2003). Can noise induce chaos?. Oikos, 102(2), 329-339.
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  Abstract: An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.
• Edmunds, J., Cushing, J. M., Costantino, R. F., Henson, S. M., Dennis, B., & Desharnais, R. A. (2003). Park's Tribolium competition experiments: A non-equilibrium species coexistence hypothesis. Journal of Animal Ecology, 72(5), 703-712.
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  Abstract: 1. In this journal 35 years ago, P. H. Leslie, T. Park and D. B. Mertz reported competitive exclusion data for two Tribolium species. It is less well-known that they also reported 'difficult to interpret' coexistence data. We suggest that the species exclusion and the species coexistence are consequences of a stable coexistence two-cycle in the presence of two stable competitive exclusion equilibria. 2. A stage-structured insect population model for two interacting species forecasts that as interspecific interaction is increased there occurs a sequence of dynamic changes (bifurcations) in which the classic Lotka-Volterra-type scenario with two stable competitive exclusion equilibria is altered abruptly to a novel scenario with three locally stable entities; namely, two competitive exclusion equilibria and a stable coexistence cycle. This scenario is novel in that it predicts the competitive coexistence of two nearly identical species on a single limiting resource and does so under circumstances of increased interspecific competition. This prediction is in contradiction to classical tenets of competition theory.
• Henson, S. M., King, A. A., Costantino, R. F., Cushing, J. M., Dennis, B., & Desharnais, R. A. (2003). Explaining and predicting patterns in stochastic population systems. Proceedings of the Royal Society B: Biological Sciences, 270(1524), 1549-1553.
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  PMID: 12908974;PMCID: PMC1691414;Abstract: Lattice effects in ecological time-series are patterns that arise because of the inherent discreteness of animal numbers. In this paper, we suggest a systematic approach for predicting lattice effects. We also show that an explanation of all the patterns in a population time-series may require more than one deterministic model, especially when the dynamics are complex.
• Henson, S. M., Reilly, J. R., Robertson, S. L., Schu, M. C., W., E., & Cushing, J. M. (2003). Predicting irregularities in population cycles. SIAM Journal on Applied Dynamical Systems, 2(2), 238-253.
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  Abstract: Oscillating population data often exhibit cycle irregularities such as episodes of damped oscillation and abrupt changes of cycle phase. The prediction of such irregularities is of interest in applications ranging from food production to wildlife management. We use concepts from dynamical systems theory to present a model-based method for quantifying the risk of impending cycle irregularity.
• Jang, S. R., & Cushing, J. M. (2003). A discrete hierarchical model of intra-specific competition. Journal of Mathematical Analysis and Applications, 280(1), 102-122.
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  Abstract: A discrete hierarchical model with either age, size, or stage structure is derived. The resulting scalar equation for total population level is then used to study contest and scramble intra-specific competition. It is shown how equilibrium levels and resilience are related for the two different competition situations. In particular, scramble competition yields a higher population level while contest competition is more resilient if the uptake rate as a function of resource density is concave down. The conclusions are reversed if the uptake rate is concave up. © 2003 Elsevier Science (USA). All rights reserved.
• Cushing, J. M., & Henson, S. M. (2002). A periodically forced Beverton-Holt equation. Journal of Difference Equations and Applications, 8(12), 1119-1120.
• Henson, S. M., & Cushing, J. M. (2002). Geometric transient solutions of autonomous scalar maps. Journal of Difference Equations and Applications, 8(1), 61-73.
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  Abstract: Smooth autonomous scalar maps with locally asymptotically stable equilibria have families of asymptotically constant solutions which decay geometrically to the equilibria. Locally, all transients converging to the equilibria have this form.
• Henson, S. M., Costantino, R. F., Desharnais, R. A., Cushing, J. M., & Dennis, B. (2002). Basins of attraction: Population dynamics with two stable 4-cycles. Oikos, 98(1), 17-24.
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  Abstract: We use the concepts of composite maps, basins of attraction, basin switching, and saddle fly-by's to make the ecological hypothesis of the existence of multiple attractors more accessible to experimental scrutiny. Specifically, in a periodically forced insect population growth model we identify multiple attractors, namely, two locally stable 4-cycles. Using the model-predicted basins of attraction, we examine data time series from a Tribolium experiment for evidence of the multiple attractors. We conclude that the multiple attractor hypothesis together with demographic stochasticity accounts for the experimental observations.
• Cushing, J. M., Henson, S. M., Desharnais, R. A., Dennis, B., Costantino, R. F., & King, A. (2001). A chaotic attractor in ecology: Theory and experimental data. Chaos, Solitons and Fractals, 12(2), 219-234.
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  Abstract: Chaos has now been documented in a laboratory population. In controlled laboratory experiments, cultures of flour beetles (Tribolium castaneum) undergo bifurcations in their dynamics as demographic parameters are manipulated. These bifurcations, including a specific route to chaos, are predicted by a well-validated deterministic model called the "LPA model". The LPA model is based on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and adult). A stochastic version of the model accounts for the deviations of data from the deterministic model and provides the means for parameterization and rigorous statistical validation. The chaotic attractor of the deterministic LPA model and the stationary distribution of the stochastic LPA model describe the experimental data in phase space with striking accuracy. In addition, model-predicted temporal patterns on the attractor are observed in the data. This paper gives a brief account of the interdisciplinary effort that obtained these results. ©2000 Elsevier Science Ltd. All rights reserved.
• Dennis, B., Desharnais, R. A., Cushing, J. M., Henson, S. M., & Costantino, R. F. (2001). Estimating chaos and complex dynamics in an insect population. Ecological Monographs, 71(2), 277-303.
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  Abstract: A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, nonlinear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attributing the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a model-predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic variability and obtained conditional least-squares estimates for the model parameters. We generated 2000 "bootstrapped data Sets" with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distributions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, including treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.
• Desharnais, R. A., Costantino, R. F., Cushing, J. M., Henson, S. M., & Dennis, B. (2001). Chaos and population control of insect outbreaks. Ecology Letters, 4(3), 229-235.
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  Abstract: We used small perturbations in adult numbers to control large fluctuations in the chaotic demographic dynamics of laboratory populations of the flour beetle Tribolium castaneum. A nonlinear mathematical model was used to identify a sensitive region of phase space where the addition of a few adult insects would result in a dampening of the life stage fluctuations. Three experimental treatments were applied: one in which perturbations were made whenever the populations were inside the sensitive region ("in-box treatment"), another where perturbations were made whenever the populations were outside the sensitive region ("out-box treatment"), and an unperturbed control. The in-box treatment caused a stabilization of insect densities at numbers well below the peak values exhibited by the out-box and control populations. This study demonstrates how small perturbations can be used to influence the chaotic dynamics of an ecological system.
• Henson, S. M., Costantino, R. F., Cushing, J. M., Desharnais, R. A., Dennis, B., & King, A. A. (2001). Lattice effects observed in chaotic dynamics of experimental populations. Science, 294(5542), 602-605.
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  PMID: 11641500;Abstract: Animals and many plants are counted in discrete units. The collection of possible values (state space) of population numbers is thus a nonnegative integer lattice. Despite this fact, many mathematical population models assume a continuum of system states. The complex dynamics, such as chaos, often displayed by such continuous-state models have stimulated much ecological research; yet discrete-state models with bounded population size can display only cyclic behavior. Motivated by data from a population experiment, we compared the predictions of discrete-state and continuous-state population models. Neither the discrete- nor continuous-state models completely account for the data. Rather, the observed dynamics are explained by a stochastic blending of the chaotic dynamics predicted by the continuous-state model and the cyclic dynamics predicted by the discrete-state models. We suggest that such lattice effects could be an important component of natural population fluctuations.
• Henson, S. M., Costantino, R. F., Cushing, J. M., Dennis, B., & Desharnais, R. A. (1999). Multiple attractors, saddles, and population dynamics in periodic habitats. Bulletin of Mathematical Biology, 61(6), 1121-1149.
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  PMID: 17879873;Abstract: Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage- structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable 'saddle cycle' separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.
• Costantino, R. F., Cushing, J. M., Dennis, B., Desharnais, R. A., & Henson, S. M. (1998). Resonant population cycles in temporally fluctuating habitats. Bulletin of Mathematical Biology, 60(2), 247-273.
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  PMID: 9559577;Abstract: Experiments with the flour beetle Tribolium have revealed that animal numbers were larger in cultures grown in a periodically fluctuating volume of medium than in cultures grown in a constant volume of the same average size. In this paper we derive and analyze a discrete stage-structured mathematical model that explains this phenomenon as a kind of resonance effect. Habitat volume is incorporated into the model by the assumption that all rates of cannibalism (larvae on eggs, adults on eggs and pupae) are inversely proportional to the volume of the culture medium. We tested this modeling assumption by conducting and statistically analyzing laboratory experiments. For parameter estimates derived from experimental data, our model indeed predicts, under certain circumstances, a larger (cycle-average) total population abundance when the habitat volume periodically fluctuates than when the habitat volume is held constant at the average volume. The model also correctly predicts certain phase relationships and transient dynamics observed in data. The analyses involve a through integration of mathematics, statistical methods, biological details and experimental data.
• Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A., & Henson, S. M. (1998). Nonlinear population dynamics: Models, experiments and data. Journal of Theoretical Biology, 194(1), 1-9.
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  PMID: 9778421;
• Cushing, J. M., Dennis, B., Desharnais, R. A., & Costantino, R. F. (1998). Moving toward an unstable equilibrium: Saddle nodes in population systems. Journal of Animal Ecology, 67(2), 298-306.
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  Abstract: 1. We identify an unstable equilibrium with a two-dimensional stable manifold and a one-dimensional unstable manifold in a three-state variable (larva, pupa, adult) insect population growth model. 2. The saddle node forecasts that the time series of some initial numbers of larvae, pupae and adults are drawn closely to the unstable equilibrium before approaching the asymptotic stable attractor (a two-cycle), while the time series of other initial points are not. 3. Using two quantitative indices, we examine time series from a Tribolium experiment for evidence of the predicted saddle node. We conclude that a saddle node accounts for the transient dynamics in these data and for the differences between the transient behaviour of different replicates of the same experiment.
• Henson, S. M., Cushing, J. M., Costantino, R. F., Dennis, B., & Desharnais, R. A. (1998). Phase switching in population cycles. Proceedings of the Royal Society B: Biological Sciences, 265(1411), 2229-2234.
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  Abstract: Oscillatory populations may exhibit a phase change in which, for example, a high-low periodic pattern switches to a low-high pattern. We propose that phase shifts correspond to stochastic jumps between basins of attraction in an appropriate phase space which associates the different phases of a periodic cycle with distinct attractors. This mechanism accounts for two-cycle phase shifts and the occurrence of asynchronous replicates in experimental cultures of Tribolium.
• Costantino, R. F., Desharnais, R. A., Cushing, J. M., & Dennis, B. (1997). Chaotic dynamics in an insect population. Science, 275(5298), 389-391.
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  Abstract: A nonlinear demographic model was used to predict the population dynamics of the flour beetle Tribolium under laboratory conditions and to establish the experimental protocol that would reveal chaotic behavior. With the adult mortality rate experimentally set high, the dynamics of animal abundance changed from equilibrium to quasiperiodic cycles to chaos as adult-stage recruitment rates were experimentally manipulated. These transitions in dynamics corresponded to those predicted by the mathematical model. Phase-space graphs of the data together with the deterministic model attractors provide convincing evidence of transitions to chaos.
• Dennis, B., Desharnais, R. A., Cushing, J. M., & Costantino, R. F. (1997). Transitions in population dynamics: Equilibria to periodic cycles to aperiodic cycles. Journal of Animal Ecology, 66(5), 704-729.
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  Abstract: 1. We experimentally set adult mortality rates, μ(a), in laboratory cultures of the flour beetle Tribolium at values predicted by a biologically based, nonlinear mathematical model to place the cultures in regions of different asymptotic dynamics. 2. Analyses of time-series residuals indicated that the stochastic stage-structured model described the data quite well. Using the model and maximum-likelihood parameter estimates, stability boundaries and bifurcation diagrams were calculated for two genetic strains. 3. The predicted transitions in dynamics were observed in the experimental cultures. The parameter estimates placed the control and μ(a) = 0.04 treatments in the region of stable equilibria. As adult mortality was increased, there was a transition in the dynamics. At μ(a) = 0.27 and 0.50 the populations were located in the two-cycle region. With μ(a) = 0.73 one genetic strain was close to a two-cycle boundary while the other strain underwent another transition and was in a region of equilibrium. In the μ(a) = 0.96 treatment both strains were close to the boundary at which a bifurcation to aperiodicities occurs; one strain was just outside this boundary, the other just inside the boundary. 4. The rigorous statistical verification of the predicted shifts in dynamical behaviour provides convincing evidence for the relevance of nonlinear mathematics in population biology.
• Henson, S. M., & Cushing, J. M. (1997). The effect of periodic habitat fluctuations on a nonlinear insect population model. Journal of Mathematical Biology, 36(2), 201-226.
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  Abstract: Laboratory data show that populations of flour beetles (Tribolium), when grown in a periodically fluctuating volume of flour, can exhibit significant increases in numbers above those attained when grown in a constant volume (of the same average). To analyze and explain this phenomenon a discrete stage-structured model of Tribolium dynamics with periodic environmental forcing is introduced and studied. This model is an appropriately modified version of an experimentally validated model for flour beetle populations growing in a constant volume of flour, in which cannibalism rates are assumed inversely proportional to flour volume. This modeling assumption has been confirmed by laboratory experiments. Theorems implying the existence and stability of periodic solutions of the periodically forced model are proved. The time averages of periodic solutions of the forced model are compared with the equilibrium levels of the unforced model (with the same average flour volume). Parameter constraints are determined for which the average population numbers in the periodic environment are greater than (or less than) the equilibrium population numbers in the associated constant environment. Sample parameter estimates taken from the literature show that these constraints are fulfilled. These theoretical results provide an explanation for the experimentally observed increase in flour beetle numbers as a result of periodically fluctuating flour volumes. More generally, these integrated theoretical and experimental results provide the first convincing example illustrating the possibility of increased population numbers in a periodically fluctuating environment. © Springer-Verlag 1997.
• Perry, J. N., Woiwod, I. P., Smith, R. N., Morse, D., Desharnais, R. A., Costantino, R. F., Cushing, J. M., & Dennis, B. (1997). Estimating chaos in an insect population. Science, 276(5320), 1881-1882.
• Cushing, J. M., Dennis, B., Desharnais, R. A., & Costantino, R. F. (1996). An interdisciplinary approach to understanding nonlinear ecological dynamics. Ecological Modelling, 92(2-3), 111-119.
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  Abstract: We describe a research program which covers a spectrum of activities essential to testing nonlinear population theory: from the translation of the biology into the formal language of mathematics, to the analysis of mathematical models, to the development and application of statistical techniques for the analysis of data, to the design and implementation of biological experiments. The statistical analyses, mathematics, and biology are thoroughly integrated. We review several aspects of our current research effort that demonstrate this integration.
• Henson, S. M., & Gushing, J. M. (1996). Hierarchical models of intra-specific competition: Scramble versus contest. Journal of Mathematical Biology, 34(7), 755-772.
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  Abstract: Hierarchical structured models for scramble and contest intra-specific competition are derived. The dynamical consequences of the two modes of competition are studied under the assumption that both populations divide up the same amount of a limiting resource at equal population levels. A comparison of equilibrium levels and their resiliences is made in order to determine which mode of competition is more advantageous. It is found that the concavity of the resource uptake rate is an important determining factor. Under certain circumstances contest competition is more advantageous for a population while under other circumstances scramble competition is more advantageous. © Springer-Verlag 1996.
• Constantino, R. F., Cushing, J. M., Dennis, B., & Desharnais, R. A. (1995). Experimentally induced transitions in the dynamic behaviour of insect populations. Nature, 375(6528), 227-230.
• Dennis, B., Desharnais, R. A., Cushing, J. M., & Costantino, R. F. (1995). Nonlinear demographic dynamics: Mathematical models, statistical methods, and biological experiments. Ecological Monographs, 65(3), 261-281.
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  Abstract: Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identities boundaries in parameter space where stable equilibria bifurcate to periodic 2-cycles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the Hour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to understanding nonlinear ecological dynamics.
• Crowe, K. M., & Cushing, J. M. (1994). Optimal instar parasitization in a stage structured host-parasitoid model. Natural Resource Modeling, 8(2), 119-138.
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  Abstract: A stage structured host-parasitoid model is derived and the equilibria studied. It is shown under what conditions the parasitoid controls an exponentially growing host in the sense that a coexistence equilibrium exists. Furthermore, for host populations whose inherent growth rate is not too large it is proved that in order to minimize the adult host equilibrium level it is necessary that the parasitoids attack only one of the larval stages. It is also proved in this case that the minimum adult host equilibrium level is attained when the parasitoids attack that larval stage which also maximizes the expected number of emerging adult parasitoid per larva at equilibrium. Numerical simulations tentatively indicate that the first conclusion remains in general valid for the model. -from Authors
• Cushing, J. M. (1994). Oscillations in age-structured population models with an Allee effect. Journal of Computational and Applied Mathematics, 52(1-3), 71-80.
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  Abstract: A nonautonomous ordinary differential delay equation for the birth rate of a model age-structured population is derived under the assumptions that nonlinear density effects on fertility exhibit an "Allee effect". It is shown how this assumption produces an interval of inherent net reproductive numbers less than one on which there exist two stable (asymptotic) equilibria. Furthermore, in the presence of a maturation delay and a sufficiently narrow age-specific fertility window, numerical solutions show that a certain type of attracting, large amplitude "synchronous" oscillation can also exist on this interval. A heuristic argument is given for the existence of such oscillations using the model obtained when the length of the fertility window shrinks to zero. © 1994.
• Cushing, J. M., & Yicang, Z. (1994). The net reproductive value and stability in matrix population models. Natural Resource Modeling, 8(4), 297-333.
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  Abstract: The net reproductive value n is defined for a general discrete linear population model with a non-negative projection matrix. This number is shown to have the biological interpretation of the expected number of offspring per individual over its life time. The main result relates n to the population's growth rate (ie the dominance eigenvalue λ of the projection matrix) and shows that the stability of the extinction state (the trivial equilibrium) can be determined by whether n is less than or greater than 1. Examples are given to show that explicit algebraic formulas for n are often derivable, and hence available for both numerical and parameter studies of stability, when no such formulas for λ are available. -Authors
• Cushing, J. M. (1992). A discrete model for competing stage-structured species. Theoretical Population Biology, 41(3), 372-387.
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  Abstract: This paper deals with the problem of relating physiological properties of individual organisms to the dynamics at the total population level. A general nonlinear matrix difference equation is described which accounts for the dynamics of stage-structured populations under the assumption that individuals in the populations can be placed into well defined descriptive stages. Density feedback is modeled through an assumption that (stage-specific) fertilities and transitions are proportional to a resource uptake functional which is dependent upon a total weighted population size. It is shown how, if stage-specific differences in mortality are insignificant compared to stage-specific differences in fertility and inter-stage transitions, a nonlinear version of the strong ergodic theorem of demography mathematically separates the population level dynamics from the dynamics of the stage distribution vector, which is shown to stabilize independently of the population level dynamics. The nonlinear dynamics at the population level are governed by a key parameter π that encapsulates the stage-specific parameters and thereby affords a means by which population level dynamics can be linked to properties of individual organisms. The method is applied to a community of stagestructured populations competing for a common limiting resource, and it is seen how the parameter π determines the competitively superior species. An example of size structured competitors illustrates how the method can relate the competitive success of a species to such size-specific properties as resource conversion efficiencies and allocation fractions for individual growth and reproduction, largest adult body size, and size at birth and maturation. © 1992.
• Cushing, J. M. (1992). A size-structured model for cannibalism. Theoretical Population Biology, 42(3), 347-361.
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  Abstract: A size-structured model for the dynamics of a cannibalistic population is derived under the assumption that cannibals (successfully) attack only smaller bodied victims, as is generally the case in the biological world. In addition to the resulting size-dependent death rate, the model incorporates the positive feedback mechanism resulting from the added resource energy obtained by the cannibal from the consumption of the victim. From the nonlinear partial integro-differential equation model, it is shown how to obtain a complete analysis of the global dynamics of the total population biomass. This analysis yields many dynamical features that have been attributed to cannibalism in the literature, including density self-regulation, a "life-boat strategy" phenomenon by which a population avoids extinction by practicing cannibalism under circumstances when it would otherwise go extinct, and multiple stable positive equilibrium states and hysteresis. © 1992.
• Cushing, J. M., & Jia, L. i. (1992). Intra-specific competition and density dependent juvenile growth. Bulletin of Mathematical Biology, 54(4), 503-519.
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  Abstract: A difference equation model for the dynamics of a semelparous size-structured species consisting of juvenile and adult individuals is derived and studied. The adult population consists of two size classes, a smaller class and a larger more fertile class. Negative feedback occurs through slowed juvenile growth due to increased total population levels during the developmental period and consequently a smaller adult size at maturation. Intra-specific competition coefficients are size dependent and measure the strength of intra-specific competition between juveniles and adults. It is shown that equilibrium states in which adults and juveniles occur together at all times are in general destabilized by significantly increased juvenile vs adults competition with the result that stable periodic cycles appear, in which the generations alternate in time and hence avoid competition. This result supports the tenet that intra-specific competition between juveniles and adults is destabilizing. Exceptions to this destabilization principle are found, however, in which populations exhibiting non-equilibrium, aperiodic dynamics can be equilibrated by increase competition between juveniles and adults. This occurs, for example, when adult fertility and competition coefficients are significantly size class dependent. © 1992 Society for Mathematical Biology.
• Cushing, J. M. (1991). A simple model of cannibalism. Mathematical Biosciences, 107(1), 47-71.
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  PMID: 1806108;Abstract: A simple nonlinear discrete model is derived for the dynamics of a two-age class population consisting of juveniles and adults that includes cannibalism of juveniles by adults. The model is investigated analytically and numerically. It is shown how even this very simple model, by incorporating the negative and positive feedbacks due to cannibalism, can account for several important phenomena concerning the dynamics of cannibalistic populations that have been discussed and studied in the literature. These include the possibilities that the practice of cannibalism can (1) in certain circumstances be a form of self-regulation that promotes stable equilibration, while in other circumstances it can lead to population oscillations; (2) result in a viable population in circumstances when its absence would otherwise result in extinction; and (3) be the source of multiple stable equilibria and hysteresis effects. © 1991.
• Cushing, J., & Cushing, J. M. (1991). A simple model of cannibalism. Mathematical biosciences, 107(1).
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  A simple nonlinear discrete model is derived for the dynamics of a two-age class population consisting of juveniles and adults that includes cannibalism of juveniles by adults. The model is investigated analytically and numerically. It is shown how even this very simple model, by incorporating the negative and positive feedbacks due to cannibalism, can account for several important phenomena concerning the dynamics of cannibalistic populations that have been discussed and studied in the literature. These include the possibilities that the practice of cannibalism can (1) in certain circumstances be a form of self-regulation that promotes stable equilibration, while in other circumstances it can lead to population oscillations; (2) result in a viable population in circumstances when its absence would otherwise result in extinction; and (3) be the source of multiple stable equilibria and hysteresis effects.
• Cushing, J. M., & Jia, L. i. (1989). On Ebenman's model for the dynamics of a population with competing juveniles and adults. Bulletin of Mathematical Biology, 51(6), 687-713.
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  Abstract: A general version of a model of Ebenman for the dynamics of a population consisting of competing juveniles and adults is analyzed using methods of bifurcation theory. A very general existence results is obtained for non-trivial equilibria and non-negative synchronous two-cycles that bifurcate simultaneously at the critical value r=1 of the inherent net reproductive rate r. Stability is studied in this general setting near the bifurcation point and conditions are derived that determine which of these two bifurcating branches is the stable branch. These general results are supplemented by numerical studies of the asymptotic dynamics over wider parameter ranges where various other bifurcations and stable attractors are found. The implications of these results are discussed with respect to the effects on stability that age class competition within a population can have and whether such competition is stabilizing or destabilizing. © 1989 Society for Mathematical Biology.
• Cushing, J. M. (1987). Equilibria and oscillations in age-structured population growth models.. Mathematical modelling of environmental and ecological systems, 153-175.
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  Abstract: Describes some recent results concerning the dynamics of some quite general age-structured population growth models and demonstrates the usefulness of a certain approach to the study of asymptotic states. This approach is based upon bifurcation theory and methods. -from Author
• Cushing, J. M. (1987). Equilibria in systems of interacting structured populations. Journal of Mathematical Biology, 24(6), 627-649.
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  PMID: 3572260;Abstract: The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k-1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given. © 1987 Springer-Verlag.
• Cushing, J. M. (1986). Oscillatory population growth in periodic environments. Theoretical Population Biology, 30(3), 289-308.
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  Abstract: A general first-order nonlinear differential equation is derived for the dynamics of a population in such a way that the inherent growth rate r and the equilibrium "carrying capacity" K appear explicitly as parameters. By means of standard regular perturbation techniques, properties of the periodic asymptotic state of the population are studied under the assumption that r and K suffer periodic perturbations of small amplitude. Specific examples are studied analytically and numerically. © 1986.
• Cushing, J. M. (1986). Periodic Lotka-Volterra competition equations. Journal of Mathematical Biology, 24(4), 381-403.
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  PMID: 3805902;Abstract: The Lotka-Volterra competition equations with periodic coefficients derived from the MacArthur-Levins theory of a one-dimensional resource niche are studied when the parameters are allowed to oscillate periodically in time. Specifically, niche positions and widths, resource availability and resource consumption rates are allowed small amplitude periodicities around a specified mean value. Two opposite cases are studied both analytically and numerically. First only resource consumption rates are allowed to oscillate while niche dimensions and resource availability are held constant. The resulting oscillations in population densities and the strength of the system stability as they depend upon crucial relative phase and amplitude differences between the species' consumption rates are studied. This leads to a clear notion of "temporal niche" and of the effects that such oscillations can have on competitive coexistence. Secondly, all system parameters are allowed to oscillate, although the oscillatory consumption rates are assumed identical for both species. The effects on the population density oscillations and their averages are studied and the "best" choice of the common, periodic resource consumption rate for these two "identical" species competing for similar (even identical) niches is considered. © 1986 Springer-Verlag.
• Cushing, J. M. (1986). Periodic McKendrick equations for age-structered population growth. Computers and Mathematics with Applications, 12(4-5 PART A), 513-526.
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  Abstract: With the averaged net reproductive rate used as a bifurcation parameter, the existence of a local parameterized branch of time-periodic solutions of the McKendrick equations is proved under the assumption that the death and fertility rates suffers small-amplitude time periodicities. The required linear theory is developed and the results are illustrated by means of a simple example in which fertility varies cosinusoidally in time. © 1986.
• Cushing, J. M. (1985). Equilibria in structured populations. Journal of Mathematical Biology, 23(1), 15-39.
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  PMID: 4078497;Abstract: The existence of a stable positive equilibrium state for the density ρ of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ρ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature. © 1985 Springer-Verlag.
• Cushing, J. M. (1985). Global branches of equilibrium solutions of the McKendrick equations for age-structured population growth. Computers and Mathematics with Applications, 11(1-3), 175-188.
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  Abstract: The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. Under only continuity assumptions on the density dependent death and tertility rates, it is shown that a global continuum of positive equilibria exists in a certain Banach space. This continuum connects from the trivial solution at n = 1 to the boundary of the domain on which the problem is posed. Results concerning the spectrum are given. In particular, some circumstances are described under which positive equilibria exist for all n values greater than the critical value n = 1. © 1985.
• Cushing, J. M. (1984). Existence and stability of equilibria in age-structured population dynamics. Journal of Mathematical Biology, 20(3), 259-276.
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  Abstract: The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results. © 1984 Springer-Verlag.
• Cushing, J. M. (1983). Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics. Computers and Mathematics with Applications, 9(3), 459-478.
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  Abstract: A local two parameter bifurcation theorem concerning the bifurcation from steady states of time periodic solutions of a nonlinear system of partial, integro-differential equations is proved. A Hopf bifurcation theorem is derived as a corollary. By means of independent and dependent variable changes this theorem is applicable to the general McKendrick equations governing the growth of an age-structured population (with the added feature here of a possible gestation period). The theorem is based on a Fredholm theory developed in the paper for the associated linear equations. An application is given to an age-structured population whose fecundity is density and age dependent and it is shown that for a sufficiently narrow age-specific "reproductive and resource consumption window" steady state instabilities, accompanied by sustained time periodic oscillations, occur when the birth modulus surpasses a critical value. © 1983.
• Cushing, J. M., & Saleem, M. (1982). A predator prey model with age structure. Journal of Mathematical Biology, 14(2), 231-250.
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  PMID: 7119585;Abstract: A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects its fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and destabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation period m of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for small m or are destabilized as m decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities. © 1982 Springer-Verlag.
• Cushing, J. M. (1980). Model stability and instability in age structured populations. Journal of Theoretical Biology, 86(4), 709-730.
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  PMID: 7253668;Abstract: Primary sources of instability and oscillation in the dynamics of a single species are various mechanisms which cause delayed responses of fecundity to environmental changes. Fundamental mechanisms include gestation periods, age-specific reproductive rates and age-specific (or weighted) effects of population density on fecundity. The effects on the stability and instability of equilibria caused by these three specific mechanisms are studied by means of a general model equation.
• Cushing, J. M. (1980). Two species competition in a periodic environment. Journal of Mathematical Biology, 10(4), 385-400.
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  Abstract: The classical Lotka-Volterra equations for two competing species have constant coefficients. In this paper these equations are studied under the assumption that the coefficients are periodic functions of a common period. As a generalization of the existence theory for equilibria in the constant coefficient case, it is shown that there exists a branch of positive periodic solutions which connects (i.e. bifurcates from) the two nontrivial periodic solutions lying on the coordinate axes. This branch exists for a finite interval or "spectrum" of bifurcation parameter values (the bifurcation parameter being the average of the net inherent growth rate of one species). The stability of these periodic solutions is studied and is related to the theory of competitive exclusion. A specific example of independent ecological interest is examined by means of which it is shown under what circumstances two species, which could not coexist in a constant environment, can coexist in a limit cycle fashion when subjected to suitable periodic harvesting or removal rates. © 1980 Springer-Verlag.
• Cushing, J. M. (1979). Stability and instability in predator-prey models with growth rate response delays. Rocky Mountain Journal of Mathematics, 9(1), 43-50.
• Cushing, J. M. (1978). Bifurcation of periodic oscillations due to delays in single species growth models. Journal of Mathematical Biology, 6(2), 145-161.
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  Abstract: Theorems are given which guarantee the bifurcation of non-constant, periodic solutions (of fixed period) of a scalar functional equation with two independent parameters. These results are applied to a single, isolated species growth model of general form with a general Volterra (Stieltjes) delay using the 'magnitudes' of the instantaneous and delayed growth rate responses as the independent bifurcation parameters. The case of linear growth rate responses (i.e. delay logistic models) is considered in more detail, particularly the often studied single lag logistic equation. © 1978 Springer-Verlag.
• Cushing, J. M. (1977). On the oscillatory nature of solutions of general predator-prey models with time delays. Nonlinear Analysis, 1(6), 583-592.
• Cushing, J. M. (1977). PERIODIC TIME-DEPENDENT PREDATOR-PREY SYSTEMS.. SIAM Journal on Applied Mathematics, 32(1), 82-95.
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  Abstract: The general system of differential equations describing predator-prey dynamics is modified by the assumption that the coefficients are periodic functions of time. By use of standard techniques of bifuraction theory, as well as a recent global result of P. H. Rabinowitz, it is shown that this system has a periodic solution (in place of an equilibrium) provided the long term time average of the predator's net, uninhibited death rate is in a suitable range. The bifurcation is from the periodic solution of the time-dependent logistic equation for the prey (which results in the absence of any predator). Numerical results which clearly show this bifurcation phenomenon are briefly discussed.
• Cushing, J. M. (1977). Stable positive periodic solutions of the time-dependent logistic equation under possible hereditary influences. Journal of Mathematical Analysis and Applications, 60(3), 747-754.
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  Abstract: The logistic equation, generalized to include time-dependent but periodic coefficients and a functional, hereditary interaction term, is shown to have a positive periodic solution provided the time-dependent net birth rate has a positive average. Under more restrictive conditions on the interaction term and the net birth rate, this solution is shown to be uniformly asymptotically stable. The approach is to treat the problem as one of the bifurcation of nontrivial positive solutions from the identically zero solution using, roughly speaking, the average of the net birth rate as a nonlinear eigenvalue. © 1977.
• Cushing, J. M. (1977). Time delays in single species growth models. Journal of Mathematical Biology, 4(3), 257-264.
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  PMID: 894151;Abstract: A general model is considered for the growth of a single species population which describes the per unit growth rate as a general functional of past population sizes. Solutions near equilibrium are studied as functions of ε = 1/b, the reciprocal of the inherent per unit growth rate b of the population in the absense of any density constraints. Roughly speaking, it is shown that for large ε the equilibrium is asymptotically stable and that for ε small the solutions show divergent oscillations around the equilibrium. In the latter case a first order approximation is obtained by means of singular perturbation methods. The results are illustrated by means of a numerically integrated delay-logistic model. © 1977 Springer-Verlag.
• Cushing, J. (1976). Bounded solutions of perturbed Volterra integrodifferential systems. Journal of Differential Equations, 20(1), 61-70.
• Cushing, J. M. (1976). FORCED ASYMPTOTICALLY PERIODIC SOLUTIONS OF PREDATOR-PREY SYSTEMS WITH OR WITHOUT HEREDITARY EFFECTS.. SIAM Journal on Applied Mathematics, 30(4), 665-674.
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  Abstract: V. Volterra's two-species predator-prey integro-differential model, which describes the dynamics of predator-prey interactions when continuously accumulating lag effects are considered, is modified by the addition of forcing or control functions. It is shown that for appropriate choices of these control functions the model possesses infinitely many asymptotically periodic solutions for small logistic loads and hereditary (or lag) effects. For certain special cases the existence of periodic solutions is proved.
• Cushing, J. M. (1976). Periodic solutions of two species interaction models with lags. Mathematical Biosciences, 31(1-2), 143-156.
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  Abstract: The classical Volterra Lotka system of differential equations modeling the interaction of two species is considered under the assumption that the interaction and/or density inhibition terms may possess time lags (of a very general nature). Under the assumption that there exists a positive equilibrium, sufficient conditions are given which guarantee the existence of nonconstant, positive periodic solution. These conditions are algebraic in nature and are easily and straightforwardly applied. A few specific examples (namely, some predator prey and competing species models) are discussed and are used to illustrate how lags are able to alter significantly the general dynamics of the classical equations. The mathematical approach is to construct periodic solutions which bifurcate from the equilibrium (using the net birth rates as free parameters) by making use of Liapunov Schmidt type expansions, an approach which requires the proof of a Fredholm type alternative for Stieltjes integrodifferential systems.
• Cushing, J. M. (1976). Predator prey interactions with time delays. Journal of Mathematical Biology, 3(3-4), 369-380.
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  PMID: 1035612;Abstract: A general (Volterra-Lotka type) integrodifferential system which describes a predator-prey interaction subject to delay effects is considered. A rather complete picture is drawn of certain qualitative aspects of the solutions as they are functions of the parameters in the system. Namely, it is argued that such systems have, roughly speaking, the following features. If the carrying capacity of the prey is smaller than a critical value then the predator goes extinct while the prey tends to this carrying capacity; and if the carrying capacity is greater than, but close to this critical value then there is a (globally) asymptotically stable positive equilibrium. However, unlike the classical, non-delay Volterra-Lotka model, if the carrying capacity of the prey is too large then this equilibrium becomes unstable. In this event there are critical values of the birth and death rates of the prey and predator respectively (which hitherto have been fixed) at which "stable" periodic solutions bifurcate from the equilibrium and hence at which the system is stabilized. These features are illustrated by means of a numerically solved example. © 1976 Springer-Verlag.
• Cushing, J. M. (1976). Stable limit cycles of time dependent multispecies interactions. Mathematical Biosciences, 31(3-4), 259-273.
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  Abstract: The standard differential system which models the interaction of n species is considered under the assumption that the coefficients (i.e., the net birth rates, the self-inhibition coefficients and the interaction coefficients) are all periodic functions of time. Conditions are given which guarantee the existence of a stable periodic limit cycle. The basic result implies, roughly, that if an n - 1 species subcommunity with a stable periodic limit cycle exists and if the interactions of the system are sufficiently weak, then the addition of the nth species will result in a stable periodic limit cycle provided its average net birth rate is larger than and close to a specified critical value; on the other hand, if this average is less than but close to the critical value, then the nth species will not survive and the system will stabilize on the limit cycle of the subcommunity. Starting with basic results concerning periodic solutions of the one species case, we apply our basic result in a "bootstrapping" manner to derive a corollary which, roughly speaking, states that if at least one species has a positive average net birth rate and if the interactions are sufficiently weak, then a multispecies community will have a stable, positive limit cycle provided that the average net birth rates of the remaining species lie in certain specified ranges. © 1976.
• Bownds, J. M., & Cushing, J. M. (1975). On preserving stability of Volterra integral equations under a general class of perturbations. Mathematical Systems Theory, 9(2), 117-131.
• Bownds, J. M., & Cushing, J. M. (1975). On the behavior of solutions of predator-prey equations with hereditary terms. Mathematical Biosciences, 26(1-2), 41-54.
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  Abstract: Some global properties of solutions of the classical integrodifferential systems, introduced by Volterra in his study of two species predator-prey populations, are studied. It is shown for large logistic loads that the predator goes to extinction and the prey tends to its carrying capacity. By use of a nonlinear approximation it is shown that for smaller logistic loads a "critical point" is asymptotically stable, while for sufficiently small logistic loads this point is unstable. These cases are demonstrated numerically for the original integrodifferential system using parameters which were computed on the basis of experimental data of S. Utida for bean-weevil vs. braconid-wasp interactions. Moreover, numerical solutions suggest further varied behavior of solutions of this system. © 1975.
• Bownds, J. M., & Cushing, J. M. (1975). SOME STABILITY THEOREMS FOR SYSTEMS OF VOLTERRA INTEGRAL EQUATIONS.. Appl Anal, 5(1), 65-77.
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  Abstract: Perturbed Volterra integral equations (which are a generalization of the initial value problem for ordinary differential equations) are studied. As a natural generalization of these concepts for ordinary differential equations, stability, uniform stability, and asymptotic stability are defined for integral equations and various theorems are proved for linear and perturbed integral equations.
• Cushing, J. (1975). Stability of perturbed Volterra integral equations. Journal of Mathematical Analysis and Applications, 50(2), 325-340.
• Cushing, J. M. (1975). Extremal tests for scalar functions of several real variables at degenerate critical points. Aequationes Mathematicae, 12(1), 109-110.
• Cushing, J. M. (1975). Extremal tests for scalar functions of several real variables at degenerate critical points. Aequationes Mathematicae, 13(1-2), 89-96.
• Bownds, J. M., & Cushing, J. M. (1973). On strong stability for linear integral equations. Mathematical Systems Theory, 7(3), 193-200.
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  Abstract: The notion of strong or adjoint stability for linear ordinary differential equations is generalized to the theory of Volterra integral equations. It is found that this generalization is not unique in that equivalent definitions for differential equations lead to different stabilities for integral equations in general. Three types of stabilities arising naturally are introduced: strong stability, adjoint stability, and uniform adjoint stability. Necessary and sufficient conditions relative to the fundamental matrix for these stabilities are proved. Some lemmas dealing with non-oscillation of solutions and a semi-group property of the fundamental matrix are also given. © 1973 Springer-Verlag New York Inc.
• Cushing, J. (1973). Nonlinear Steklov problems on nonsymmetric domains. Journal of Mathematical Analysis and Applications, 43(3), 743-753.
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  Abstract: A way of formulating nonlinear Steklov problems on nonsymmetric domains as an operator equation u = μPu, where P is completely continuous, is given. Local and global existence theorems then follow from standard techniques; these results extend earlier results for symmetric domains and equations with symmetric coefficients. Some miscellaneous results are given concerning the nature of the solution branches. © 1973.
• Cushing, J. M. (1973). Strong stability of perturbed systems of Volterra integral equations. Mathematical Systems Theory, 7(4), 360-366.
• Cushing, J. (1972). Nonlinear Steklov problems on the unit circle. Journal of Mathematical Analysis and Applications, 38(3), 766-783.
• Cushing, J. (1972). Nonlinear Steklov problems on the unit circle. II (and a hydrodynamical application). Journal of Mathematical Analysis and Applications, 39(2), 267-278.
• Cushing, J. (1971). A uniqueness criterion for harmonic functions under nonlinear boundary conditions. Journal of Mathematical Analysis and Applications, 33(2), 443-448.
• Cushing, J. M. (1971). Some existence theorems for nonlinear eigenvalue problems associated with elliptic equations. Archive for Rational Mechanics and Analysis, 42(1), 63-76.
• Cushing, J., & Dunninger, D. (1971). Uniqueness theorems under nonlinear boundary conditions for some quasi-linear elliptic equations. Journal of Mathematical Analysis and Applications, 34(1), 114-127.
• Cushing, J. M. (1970). Local uniqueness for harmonic functions under nonlinear boundary conditions. Journal of Differential Equations, 8(3), 512-522.
• Cushing, J. M. (1969). Uniqueness and comparison of harmonic functions under nonlinear boundary conditions. Journal of Mathematical Analysis and Applications, 28(3), 581-589.

Presentations

• Cushing, J. M. (2020, January). Darwinian Dynamic Models of Adaptation in Changing Environments. Special Session on Modeling Natural Resources, Joint Mathematical Meetings. Denver, CO: American Mathematical Society.
• Cushing, J. M. (2020, Winter). Survival or extinction: difference equation models of evolutionary adaptation. Special Session Differential and Difference Equations in Biological Dynamics, Joint Mathematical Meetings. Denver, CO: American Mathematical Society.
• Cushing, J. M. (2019, July). Backward Bifurcations in Evolutionary Models of Population Dynamics with Applications to Life History Adaptations. Colloquium talk. Xi'an Jiatong University, Xi'an, China: Xi'an Jiatong University.
• Cushing, J. M. (2019, July). Backward Bifurcations in Evolutionary Models of Population Dynamics with Applications to Life History Adaptations. Guangzhou Mathematics-Tianyuan Workshop on Mathematical Modeling and Analysis of Population Dynamics. University of Guangzhou, China: University of Guangzhou, China.
• Cushing, J. M. (2019, July). Evolution of Life History Strategies: Semelparity versus Iteroparity. Annual meeting of the Society of Mathematical Biology. Montreal, Canada: Society of Mathematical Biology.
• Cushing, J. M. (2019, July). The Fundamental Bifurcation Theorem for the Evolutionary Dynamics of Structured Populations with Applications to Life History Adaptations. Colloquium talk. Sichuan University, Chengdu, China: Sichuan University.
• Cushing, J. M. (2019, June). Difference Equation Models for Darwinian Evolution. 25th International Conference on Difference Equations and Applications. University College London, London UK: International Society of Difference Equations.
• Cushing, J. M. (2019, May). Does Evolution Favor a Semelparity-annual or an Iteroparous-perennial Life History Strategy?. Computational and Mathematical Populations Dynamics 5. Ft. Lauderdale Florida.
• Cushing, J. M. (2018, April). Periodic matrix models for seasonal dynamics of structured populations with application to a seabird population faced with climate change. Spring Western Sectional Meeting of the American Mathematical Society. Portland State University: American Mathematical Society.
• Cushing, J. M. (2018, June). Matrix Models for the Darwinian Dynamics of Structured Populations. Workshop on Ecological and Biological Systems. Institute for Mathematics and its Applications (IMA): University of Minnesota in Minneapolis.
• Cushing, J. M. (2018, May). Equilibrium bifurcations in difference equation models from structured population dynamics. International Conference on Difference Equations and Applications. Technische Universität Dresden, Dresden, Germany: International Society of Difference Equations.
• Cushing, J. M. (2018, October). Using Darwinian Difference Equation Models to Study Life History Strategies: Semelparity vs. Iteroparity. Special Session on Mathematical Biology, Regional Meeting of the American Mathematical Society. San Francisco State University: American Mathematical Society.
• Cushing, J. M. (2018, Spring). Difference equation models in population dynamics whose coefficients are subject to Darwinian evolution. Joint Mathematics Meetings, Special Session on Real Discrete Dynamical Systems with ApplicationsAmerican Mathematical Society.
• Cushing, J. M. (2018, Spring). Equilibrium bifurcations in difference equation models from structured population dynamics: extinction, survival, and strong Allee effects. Joint Mathematics Meetings, Special Session on Bifurcations of Difference Equations and Discrete Dynamical Systems. San Diego: American Mathematical Society.
• Cushing, J. M. (2018, Spring). Remarks on the definition of R0. Joint Mathematics Meetings, Special Session on Mathematical Modeling of Natural Resources. San Diego: American Mathematical Society.
• Cushing, J. M. (2017, Spring). Periodic matrix models for seasonal dynamics of stage structured populations I: A general class of models. Joint Mathematics Meetings, Special Session on Discrete Dynamical Systems and Applications. San Diego, CA: American Mathematical Society.
• Cushing, J. M. (2017, Summer). Matrix Models for Evolutionary Population Dynamics: Studies of the Effects of Climate Change on Seabirds. International Conference on Mathematical Methods and Models in Biosciences. Skukuza Camp, Kruger Park, South Africa.
• Cushing, J. M. (2017, Summer). Matrix Models for Evolutionary Population Dynamics: Studies of the Effects of Climate Change on Seabirds. SIAM 2017 Annual Meeting Minisymposium on New Developments in Population Dynamics and Epidemiology. Pittsburgh, PA: Society of Industrial and Applied Mathematics.
• Cushing, J. M. (2017, Summer). Matrix Population Models with Imprimitive Projection Matrices With Applications to Seabird Populations & Effects of Climate Change. Minisymposium on Linear Algebra and Mathematical Biology. Iowa State University, Ames: 21st conference of the International Linear Algebra Society.
• Cushing, J. M. (2016, July). Matrix Models in Structured Population Dynamics and the Fundamental Bifurcation Theorem when the Projection Matrix is Imprimitive. 22nd International Conference on Difference Equations and Applications. Osaka Prefecture University, Osaka, Japan: International Society of Difference Equations.
• Cushing, J. M. (2016, June). Some population dynamic models used to investigate effects of climate change on life history strategies. Annual Meeting of the Canadian Applied and Industrial Mathematics Society, Symposium on Recent Advances in Dynamical Systems and Mathematical Biology. University of Alberta, Edmonton, CA: Canadian Applied and Industrial Mathematics Society.
• Cushing, J. M. (2016, May). Matrix Models for Population & Evolutionary Dynamics: Climate Change Studies on Protection Island Wildlife Refuge. Biology and Medicine through Mathematics.
• Cushing, J. M. (2016, November). Darwinian dynamics and matrix models for structured populations. 2016 Roeling Applied Mathematics Conference. University of Louisiana at Lafeyette, LA.
• Cushing, J. M. (2016, july). Matrix Models for the Evolutionary Dynamics of Structured Populations. Pre-conference to the 22nd International Conference on Difference Equations and Applications. Okayama University of Science, Okayama, Japan.
• Cushing, J. M., Henson, S., & Hayward, J. (2016, June). The Effects of Climate Change on Marine Birds Part II: Population Dynamic and Evolutionary Outcomes. Annual World Conference on Natural Resource Modeling. Northern Arizona University, Flagstaff, Arizona: Natural Modeling Association.
• Cushing, J. M. (2015, Fall). A juvenile-adult population model: climate change, cannibalism, reproductive synchrony, and strong Allee effects. Biomathematics Seminar. Arizona State University.
• Cushing, J. M. (2015, Fall). On the dynamics of an evolutionary population dynamic model and life history adaptations to climate change. 5th International Conference on Mathematical Modeling and Analysis of Populations in Biological Systems. University of Western Ontario, Ontario, CA.
• Cushing, J. M. (2015, January). A model for the Darwinian dynamics of an adult‐on‐juvenile cannibalistic population. Joint Mathematics Meeting, AMS special session on Mathematics in Natural Resource Modeling. San Antonio, TX: American Mathematical Society.
• Cushing, J. M. (2015, January). Dynamic Models for Adaptive Life History Strategies in Response to Climate Change: Can Cannibalism Save the Day?. Joint Mathematics Meeting, BIO SIGMAA, Biology Special Interest Group of the MAA,. San Antonio, TX: Mathematical Association of America.
• Cushing, J. M. (2015, January). Environmental change and life history strategies: cannibalism and reproductive synchrony. Joint Mathematics Meeting, MAA session on Recent Advances on Mathematical Modeling of the Environment and Infectious Diseases. San Antonio, TX: Mathematical Association of America.
• Cushing, J. M. (2015, July). Environmental change and life history strategies: cannibalism & reproductive synchrony. Mini-symposium on Complex Adaptive and Evolutionary Models in Biology held at the conference on Mathematical Models in Ecology and Evolution. Collège de France, Paris, France.
• Cushing, J. M. (2015, July). Matrix Models in Population and Evolutionary Dynamics: The Fundamental Bifurcation Theorem. International Conference on Recent Advances in Mathematical Biology, Analysis and Applications. University of Aligarh, Aligarh, India.
• Cushing, J. M. (2015, June). Reproductive Synchrony, Cannibalism, Climate Change, and Imprimitive Matrix Models for Structured Population Dynamics. Annual Meeting of the Society of Mathematical Biology. Atalanta, GA.
• Cushing, J. M. (2015, March). Environmental change and life history strategies: cannibalism and reproductive synchrony. Special session on New Tends in Mathematical Biology, AMS Sectional Meeting, University of Alabama HuntsvilleAmerican Mathematical Society.
• Cushing, J. M. (2014, January). Cannibalism can allow survival of a population endangered by decreased environmental resource availability. Joint Mathematics Meeting, AMS Special Session on Mathematics in Natural Resource Modeling, Mathematics of Planet Earth. Baltimore, MD: American Mathematical Society.
• Cushing, J. M. (2014, January). Difference Equation Models for the Evolutionary Dynamics of Populations Subject to Strong Allee Effects. Joint Mathematics Meeting, AMS Special Session on Difference Equations and Applications. Baltimore, MD: American Mathematical Society.
• Cushing, J. M. (2014, July). Matrix models, strong Allee effects and adaptive changes in biological populations to environmental change. 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications. Madrid, Spain: American Institute of Mathematical Sciences.
• Cushing, J. M. (2014, May). Population & Evolutionary Dynamic Consequences of Climate Change: Life History Strategies & Population Survival. International Conference on Nonlinear Dynamics with Applications to Biology. Hsnschu, Taiwan: National Center for Theoretical Science.