William M Schaffer
 Professor, Ecology and Evolutionary Biology
Contact
 (520) 6217295
 Biological Sciences East, Rm. 215A
 Tucson, AZ 85721
 wms@email.arizona.edu
Biography
Born Elizabeth, N.J. Prepared Hackley School. BS. Yale College; MS, Ph.D. Princeton University. Appointments. University of Utah (19721975); University of Arizona (1975). Guggenheim Fellow (1984). 100+ publications. $1.3x10^6 lifetime extra mural funding.
Degrees
 Ph.D. Biology
 Princeton University, Princeton, New Jersey, U.S.
 Life History Evolution
 M.S. Bioology
 Princeton University, Princeton, New Jersey, United States
 B.S. Biology
 Yale University, New Haven, Connecticut, United States
Work Experience
 Professor, University of Arizona, Tucson, Arizona (1975  Ongoing)
Interests
Teaching
Introductory Biology. Evolution of Evolution with emphasis on sciencesociety interaction.
Research
Nonlinear dynamics in biology. Mathematical Modeling of Science as a Process
Courses
201819 Courses

Biology Lecture Tutor
ECOL 497B (Fall 2018) 
Discovery of Evolution
ECOL 249 (Fall 2018) 
Introductory Biology II
ECOL 182R (Fall 2018)
201718 Courses

Biology Lecture Tutor
ECOL 497B (Spring 2018) 
Introductory Biology II
ECOL 182R (Spring 2018) 
Discovery of Evolution
ECOL 249 (Fall 2017)
201617 Courses

Biology Lecture Tutor
ECOL 497B (Spring 2017) 
Introductory Biology II
ECOL 182R (Spring 2017) 
Population Biology
ECOL 596B (Spring 2017) 
Population Biology
ECOL 596B (Fall 2016)
Scholarly Contributions
Chapters
 MukaetovaLadinska, E. B., Schaffer, W. M., Bronnikova, T. V., Westwood, J., & Perry, E. K. (2012). Cholinergic therapy for autistic spectrum disorders: Review and case report. In Cholinesterase: Production, Uses and Health Effects(pp 3365).More infoAbstract: The causes of autism are heterogeneous and still largely unknown. Currently available treatments especially for the behavioural problems frequently reported in children and adults with autistic spectrum disorder (ASD) are largely symptomatic. The cholinergic abnormalities are rather consistently reported in various molecular pathological studies, both in children and adults with ASD, and they may underlie the numerous cognitive and behavioural changes seen in ASD, e.g. cognitive changes, memory problems, attentional dysfunction etc. This raises the prospect of the use of cholinesterase inhibitors and other cholinomimetics (chemicals that can act by either directly stimulating the nicotinic or muscarinic receptors, or promote acetylcholine release) in ASD for treatment of both cognitive and behavioural changes, as well as activities of daily living and improving the overall global functioning in these subjects, similar to the effect these treatments have in various neurodegenerative and neurodevelopmental conditions (e.g. dementia and schizophrenia). We provide an overview of the current use of cholinesterase inhibitors (donepezil, galantamine and rivastigmine) and cholinomimetics (e.g. nicotine) in the treatment of some of the cognitive and behavioral symptoms in ASD. In greater detail, we discuss the potential use of cholinomimetics in these subjects, and review the experience of Mr. A, a now twentyeight yearold nonsmoking male with a history of severe behavioral dysfunction who has been wearing a nicotine patch since July, 2005. © 2012 by Nova Science Publishers, Inc. All rights reserved.
Journals/Publications
 Schaffer, W. M., & Bronnikova, T. V. (2012). PeroxidaseROS interactions. Nonlinear Dynamics, 68(3), 413430.More infoAbstract: Reactive oxygen species (ROS), such as hydrogen peroxide and superoxide anion radical, have long been recognized as harmful byproducts of oxidative metabolism. Under normal physiologic conditions, hydrogen peroxide and superoxide are detoxified by antioxidant enzymes such as catalase (CAT), superoxide dismutase (SOD), and glutathione peroxidase (GPx). Heme peroxidases (eosinophil peroxidase (EPO), lactoperoxidase (LPO), myeloperoxidase (MPO), etc.) also consume ROS, but unlike scavenging enzymes, are sources of these species as well. In the present paper, we study a welltested model of the peroxidaseoxidase (PO) reaction based on horseradish peroxidase (HRP) chemistry with regard to the production and consumption of hydrogen peroxide and superoxide. Our principal results are these: 1. PO reactions can transduce continuing infusions of hydrogen peroxide and superoxide into bounded dynamics. 2. Absent exogenous ROS input, and under conditions that retard hydrogen donor autoxidation, PO reactions can manifest low frequency bursting whereby pulses of ROS are produced at clinically significant intervals. The relevance of these results to the functional significance of fluctuating ROS concentrations in vivo, to neurodevelopmental and neurodegenerative disease and to episodic and progressive symptomatology is discussed. © Springer Science+Business Media B.V. 2012.
 Schaffer, W. M., & Bronnikova, T. V. (2011). Modeling peroxidaseoxidase interactions. ASME 2011 Dynamic Systems and Control Conference and Bath/ASME Symposium on Fluid Power and Motion Control, DSCC 2011, 2, 537544.More infoAbstract: Reactive oxygen species (ROS) and peroxidaseoxidase (PO) reactions are Janusfaced contributors to cellular metabolism. At low concentrations, reactive oxygen species serve as signaling molecules; at high concentrations, as destroyers of proteins, lipids and DNA. Correspondingly, PO reactions are both sources and consumers of ROS. In the present paper, we study a welltested model of the PO reaction based on horseradish peroxidase chemistry. Our principal predictions are these: 1. Under hypoxia, the PO reaction can emit pulses of hydrogen peroxide at apparently arbitrarily long intervals. 2. For a wide range of input rates, continuing infusions of ROS are transduced into bounded dynamics. 3. The response to ROS input is hysteretic. 4. With sufficient input, regulatory capacity is exceeded and hydrogen peroxide, but not superoxide, accumulates. These results are discussed with regard to the episodic nature of neuro developmental and neurodegenerative diseases that have been linked to oxidative stress and to downstream interactions that may result in positive feedback and pathology of increasing severity. Copyright © 2011 by ASME.
 Schaffer, W. M. (2009). A surfeit of cycles. Energy and Environment, 20(6), 985996.More infoAbstract: Chaotic sets are organized about "skeletons" of periodic orbits in the sense that every point on a chaotic set is arbitrarily close to such an orbit. The orbits have the stability property of saddles: attracting in some directions; repelling in others. This topology has implications for changing climates that evidence pronounced variability on time scales ranging from decades to tens of thousands of years. Among these implications are the following: 1. A wide range of periodicities should be (and are) observed. 2. Periodicities should (and do) shift  often abruptly  as the evolving climatic trajectory sequentially shadows first one periodic orbit and then another. 3. Models that have been "tuned" (parametric adjustment) to fit trajectorial evolution in the vicinity of one periodic orbit are likely to fail when the real system moves to another region of the phase space. 4. In response to secular forcing, chaotic sets simplify via the elimination of periodic orbits. If one accepts the reality of anthropogenic warming, the longterm prediction is loss of intrinsic variability. 5. In response to periodic forcing, nonlinear systems can manifest subharmonic resonance i.e., "cyclic" behavior with periods and rotation numbers rationally related to the period of the forcing. Such cycling has been implicated in millennial and stadial variations in paleoclimatic time series. 6. Generically, the dynamics of system observables, such as climate sensitivity, are qualitatively equivalent to those of the whole. If the climate is chaotic, so too is sensitivity. These considerations receive minimal attention in consensus views of climate change that emphasize essentially onetoone correspondence between global temperatures and exogenous forcing. Caveat emptor.
 Schaffer, W. M., & Bronnikova, T. V. (2009). Controlling malaria: competition, seasonality and 'slingshotting' transgenic mosquitoes into natural populations.. Journal of biological dynamics, 3(23), 286304.More infoPMID: 22880835;Abstract: Forty years after the World Health Organization abandoned its eradication campaign, malaria remains a public health problem of the first magnitude with worldwide infection rates on the order of 300 million souls. The present paper reviews potential control strategies from the viewpoint of mathematical epidemiology. Following MacDonald and others, we argue in Section 1 that the use of imagicides, i.e., killing, or at least repelling, adult mosquitoes, is inherently the most effective way of combating the pandemic. In Section 2, we model competition between wildtype (WT) and plasmodiumresistant, genetically modified (GM) mosquitoes. Under the assumptions of inherent cost and prevalencedependant benefit to transgenics, GM introduction can never eradicate malaria save by stochastic extinction of WTs. Moreover, alternative interventions that reduce prevalence have the undesirable consequence of reducing the likelihood of successful GM introduction. Section 3 considers the possibility of using seasonal fluctuations in mosquito abundance and disease prevalence to 'slingshot' GM mosquitoes into natural populations. By introducing GM mosquitoes when natural populations are about to expand, one can 'piggyback' on the yearly cycle. Importantly, this effect is only significant when transgene cost is small, in which case the nontrivial equilibrium is a focus (damped oscillations), and piggybacking is amplified by the system's inherent tendency to oscillate. By way of contrast, when transgene cost is large, the equilibrium is a node and no such amplification is obtained.
 Schaffer, W. M. (2008). Human population and carbon dioxide. Energy Policy, 36(7), 27612764.More infoAbstract: A recently proposed model of human population and carbon utilization is reviewed. Depending on parameter values, one of three possible longterm outcomes is obtained. (1) Atmospheric carbon, (CO2)atm, and human populations equilibrate at positive values. (2) The human population stabilizes, while (CO2)atm increases without bound. (3) The human population goes extinct and atmospheric carbon declines to 0. The final possibility is qualitatively compatible with both "consensus" views of climate change and the opinions of those who are more impressed with the manifestly adverse consequences of carbonmitigation to human reproduction and survival. © 2008 Elsevier Ltd. All rights reserved.
 Schaffer, W. M., & Bronnikova, T. V. (2007). Parametric dependence in model epidemics. I: contactrelated parameters.. Journal of biological dynamics, 1(2), 183195.More infoPMID: 22873340;Abstract: One of the interesting properties of nonlinear dynamical systems is that arbitrarily small changes in parameter values can induce qualitative changes in behavior. The changes are called bifurcations, and they are typically visualized by plotting asymptotic dynamics against a parameter. In some cases, the resulting bifurcation diagram is unique: irrespective of initial conditions, the same dynamical sequence obtains. In other cases, initial conditions do matter, and there are coexisting sequences. Here we study an epidemiological model in which multiple bifurcation sequences yield to a single sequence in response to varying a second parameter. We call this simplification the emergence of unique parametric dependence (UPD) and discuss how it relates to the model's overall response to parameters. In so doing, we tie together a number of threads that have been developing since the mid1980s. These include perioddoubling; subharmonic resonance, attractor merging and subduction and the evolution of strange invariant sets. The present paper focuses on contact related parameters. A followup paper, to be published in this journal, will consider the effects of noncontact related parameters.
 Schaffer, W. M., & Bronnikova, T. V. (2007). Parametric dependence in model epidemics. II: Noncontact raterelated parameters.. Journal of biological dynamics, 1(3), 231248.More infoPMID: 22876793;Abstract: In a previous paper, we discussed the bifurcation structure of SEIR equations subject to seasonality. There, the focus was on parameters that affect transmission: the mean contact rate, β(0), and the magnitude of seasonality, ε(B). Using numerical continuation and brute force simulation, we characterized a global pattern of parametric dependence in terms of subharmonic resonances and perioddoublings of the annual cycle. In the present paper, we extend this analysis and consider the effects of varying noncontactrelated parameters: periods of latency, infection and immunity, and rates of mortality and reproduction, which, following the usual practice, are assumed to be equal. The emergence of several new forms of dynamical complexity notwithstanding, the pattern previously reported is preserved. More precisely, the principal effect of varying noncontact related parameters is to displace bifurcation curves in the β(0)ε(B) parameter plane and to expand or contract the regions of resonance and perioddoubling they delimit. Implications of this observation with respect to modeling realworld epidemics are considered.
 Olsen, L. F., Bronnikova, T. V., & Schaffer, W. M. (2002). Secondary quasiperiodicity in the peroxidaseoxidase reaction. Physical Chemistry Chemical Physics, 4(8), 12921298.More infoAbstract: Secondary quasiperiodicity (perioddoubled oscillations modulated by an incommensurate frequency), or "Q2", is the temporal manifestation of quasiperiodic motion on perioddoubled tori. The existence of this regime in a chemical reaction was first predicted (T. V. Bronnikova, W. M. Schaffer and L. F. Olsen, J. Chem. Phys., 1996, 105, 10849) in the course of numerical explorations of a detailed model of the peroxidaseoxidase system. Subsequent analysis (T. V. Bronnikova, W. M. Schaffer, M. J. B. Hauser and L. F. Olsen, J. Phys. Chem. B, 1998, 102, 632) suggested the possibility of homoclinic transitions ("fat torus" bifurcation) to chaos involving Q2. In the present paper, we present the first experimental evidence for secondary quasiperiodicity and fat torus bifurcations in a chemical oscillator. We also identify a second ("thin torus") route to chaos involving Q2. The relationship of these two bifurcation scenarios to each other and to the experimental findings is discussed.
 Bronnikova, T. V., Schaffer, W. M., & Olsen, L. F. (2001). Nonlinear dynamics of the peroxidaseoxidase reaction: I. Bistability and bursting oscillations at low enzyme concentrations. Journal of Physical Chemistry B, 105(1), 310321.More infoAbstract: Under CSTR or semibatch conditions, the horseradish peroxidase (HRP)catalyzed peroxidaseoxidase (PO) reaction evidences a wide range of nonlinear dynamical behaviors. Many of these regimes have proved to be predictable by a detailed model of the reaction first proposed in 1995. This model, which we refer to as BFSO, can also account for experimentally observed bifurcation sequences in response to varying concentrations of phenolic modifiers and rates of hydrogen donor input. Among those findings for which the model cannot account is the observation of bistability and bursting at low enzyme concentrations. This deficiency is important not only because these phenomena are biologically important but also because their existence requires a topology which, for the experimental circumstances in question, appears to be inconsistent with the model as originally formulated. In the present paper, we show that this deficiency can be remedied by the inclusion of an additional reaction whereby NADH and Superoxide anion react in the presence of hydrogen ion to produce NAD radicals and hydrogen peroxide. Comparisn of the modified model's behavior with laboratory experiments suggests semiquantitative agreement between theory and observation. In particular, the model is . able to reproduce experimentally observed responses to shortterm perturbation by oxygen input suspension and the addition of hydrogen peroxide to the reaction mixture, as well as what was first described as "autonomous" switching between stable and oscillatory dynamics. Mathematically, addition of the new reaction makes possible the interaction of Hopf and hysteresis instabilities, as previously described in the BelousovZhabotinskyi reaction. © 2001 American Chemical Society.
 King, A. A., & Schaffer, W. M. (2001). The geometry of a population cycle: A mechanistic model of snowshoe hare demography. Ecology, 82(3), 814830.More infoAbstract: The phenomenology and causes of snowshoe hare cycles are addressed via construction of a threetrophiclevel population dynamics model in which hare populations are limited by the availability of winter browse from below and by predation from above. In the absence of predators, the model predicts annual oscillations, the magnitude of which depends on habitat quality. With predators in the system, a wide range of additional dynamics are possible: multiannual cycles of various periods, quasiperiodicity, and chaos. Parameterizing the model from the literature leads to the conclusion that the model is compatible with the principal features of the cycle in nature: its regularity, mean period, and the observed range of peaktotrough amplitudes. The model also points to circumstances that can result in the cycle's abolition as observed, for example, at the southern edge of the hare's range. The model predicts that the increase phase of the cycle is brought to a halt by food limitation, while the decline from peak numbers is a consequence of predation. This is consistent with factorial field experiments in which hare populations were given Supplemental food and partial surcease from predators. The results of the experiments themselves are also reproducible by the model. Analysis of the model was carried out using a recently developed method in which the original dynamical system is reformulated as a perturbation of a Hamiltonian limit wherein exist infinite numbers of periodic, quasiperiodic, and chaotic motions. The periodic Orbits are continued numerically into nonHamiltonian regions of parameter space corresponding to the situation in nature. This procedure allows one to obtain an overall understanding of the geometry of parametric dependencies. The present study represents the first formulation of a full threetrophiclevel snowshoe hare model and the first time any model of the cycle has been parameterized entirely using independently measured quantities.
 Schaffer, W. M., Bronnikova, T. V., & Olsen, L. F. (2001). Nonlinear dynamics of the peroxidaseoxidase reaction. II Compatibility of an extended model with previously reported modeldata correspondences. Journal of Physical Chemistry B, 105(22), 53315340.More infoAbstract: In the course of formulating detailed models of complex chemical reactions, it is sometimes the case that modifications intended to account for one set of experimental observations wind up destroying a model's ability to account for other results. Here, we consider a recently proposed model of the peroxidaseoxidase reaction which derives from an earlier scheme via the addition of NADH oxidation by superoxide anion or its protonated form, hydroperoxyl radical. This modification was introduced to account for the observation of bistability and bursing at enzyme concentrations less than 0.5 μM. Left unanswered in our previous paper was the matter of whether the proposed "fix" invalidates previously published examples of modeldata agreement at higher enzyme concentrations. In the present paper, we show that under these latter circumstances, the new mechanism is as good as, and in some instances superior to, its predecessor. More generally, we argue that the consequences of NADH oxidation by O2 or HO2̇ should be manifest principally at low enzyme concentrations, thereby offering a "global" explanation of our findings. Neither our original model, nor the derivative scheme treated here, provides for reactions involving NAD dimers, a species in which there has recently been renewed interest. Most importantly, it has been proposed to replace the reduction of coIII(an enzyme intermediate) by NAD radicals, a reaction for which there is no direct evidence, with the corresponding reaction involving NAD2. While a detailed assessment of the consequences of dimer chemistry to theoretical peroxidaseoxidase dynamics is beyond the scope of the present investigation, it is easily documented that this substitution, by itself, abolishes oscillary behavior for all model parametrizations previously considered. Moreover, this result appears to obtain for arbitrarily small values of the associated rate constant. Whether or not the inclusion of additional dimer reactions can restore the model's ability to account for experimental observations of complex dynamics remains to be determined.
 Schaffer, W. M., Pederson, B. S., Moore, B. K., Skarpaas, O., King, A. A., & Bronnikova, T. V. (2001). Subharmonic resonance and multiannual oscillations in northern mammals: A nonlinear dynamical systems perspective. Chaos, solitons and fractals, 12(2), 251264.More infoAbstract: We conjecture that the wellknown oscillations (3 to 5yr and 10yr cycles) of northern mammals are examples of subharmonic resonance which obtains when ecological oscillators (predatorprey interactions) are subject to periodic forcing by the annual march of the seasons. The implications of this hypothesis are examined through analysis of a barebones, Hamiltonian model which, despite its simplicity, nonetheless exhibits the principal dynamical features of more realistic schemes. Specifically, we describe the genesis and destruction of resonant oscillations in response to variation in the intrinsic time scales of predator and prey. Our analysis suggests that cycle period should scale allometrically with body size, a fact first commented upon in the empirical literature some years ago. Our calculations further suggest that the dynamics of cyclic species should be phase coherent, i.e., that the intervals between successive maxima in the corresponding time series should be more nearly constant than their amplitude  a prediction which is also consistent with observation. We conclude by observing that complex dynamics in more realistic models can often be continued back to Hamiltonian limits of the sort here considered.
 King, A. A., & Schaffer, W. M. (1999). The rainbow bridge: Hamiltonian limits and resonance in predatorprey dynamics. Journal of Mathematical Biology, 39(5), 439469.More infoPMID: 10602912;Abstract: In the presence of seasonal forcing, the intricate topology of nonintegrable Hamiltonian predatorprey models is shown to exercise profound effects on the dynamics and bifurcation structure of more realistic schemes which do not admit a Hamiltonian formulation. The demonstration of this fact is accomplished by writing the more general models as perturbations of a Hamiltonian limit, ℋ, in which are contained infinite numbers of periodic, quasiperiodic and chaotic motions. From ℋ, there emanates a surface, Γ, of NejmarkSacker bifurcations whereby the annual oscillations induced by seasonality are destabilized. Connecting Γ and ℋ is a bridge of resonance horns within which invariant motions of the Hamiltonian case persist. The boundaries of the resonance horns are curves of tangent (saddlenode) bifurcations corresponding to subharmonics of the yearly cycle. Associated with each horn is a rotation number which determines the dominant frequency, or "color", of attractors within the horn. When viewed through the necessarily coarse filter of ecological data acquisition, and regardless of their detailed topology, these attractors are often indistinguishable from multiannual cycles. Because the tips of the horns line up monotonically along Γ, it further follows that the distribution of observable periods in systems subject to fluctuating parameter values induced, for example, by yeartoyear variations in the climate, will often exhibit a discernible central tendency. In short, the bifurcation structure is consistent with the observation of multiannual cycles in Nature. Fundamentally, this is a consequence of the fact that the bridge between ℋ and Γ is a rainbow bridge. While the present analysis is principally concerned with the two species case (one predator and one prey), Hamiltonian limits are also observed in other ecological contexts: 2nspecies (n predators, n prey) systems and periodicallyforced three level food chain models. Hamiltonian limits may thus be common in models involving the destruction of one species by another. Given the oftcommented upon structural instability of Hamiltonian systems and the corresponding lack of regard in which they are held as useful caricatures of ecological interactions, the pivotal role assigned here to Hamiltonian limits constitutes a qualitative break with the conventional wisdom. © SpringerVerlag 1999.
 Bronnikova, T. V., Schaffer, W. M., J., M., & Olsen, L. F. (1998). Routes to chaos in the peroxidaseoxidase reaction. 2. The fat torus scenario. Journal of Physical Chemistry B, 102(3), 632640.More infoAbstract: Experimental studies [Hauser, M. J. B.; Olsen, L. F. J. Chem. Soc., Faraday Trans. 1996, 92, 28572863] of the peroxidaseoxidase (PO) reaction at pH values in excess of 5.4 suggest the existence of narrow regions of complex dynamics between adjacent mixedmode oscillations (MMOs) that occur in periodadding sequences. Previously [Hauser, M. J. B.; Olsen, L. F.; Bronnikova, T. V.; Schaffer, W. M. J. Phys. Chem. B 1997, 101, 50755083], it was argued that both the periodadding sequences and the transitional regions between neighboring MMOs are predictable by a detailed model of the reaction called BFSO [Bronnikova, T. V.; Fed'kina, V. R.; Schaffer, W. M.; Olsen, L. F. J. Phys. Chem. 1995, 99, 93099312]. In the present paper, we study the transitional regions via computer simulation. Our investigations indicate that the motion therein may be periodic, quasiperiodic, or chaotic. In greater detail, we observe a quasiperiodic route to chaos whereby perioddoubled cycles give rise to doubled tori that, in turn, undergo homoclinic bifurcations to chaos. Because the latter transitions are a consequence of progressive fattening of the tori, we propose calling this scenario the "fat torus" route to chaos, and the homoclinic bifurcations "fat torus" bifurcations (FTBs). The numerical results are qualitatively consistent with the experimental findings reported to date. FTBs and the resultant perioddoubled, fractal tori may provide a criterion for discriminating among alternative models of the PO reaction.
 J., M., Olsen, L. F., Bronnikova, T. V., & Schaffer, W. M. (1997). Routes to chaos in the peroxidaseoxidase reaction: Perioddoubling and periodadding. Journal of Physical Chemistry B, 101(25), 50755083.More infoAbstract: Previous investigations of the peroxidaseoxidase reaction indicate the existence of a perioddoubling route to chaos at pH 5.2 and a periodadding route at pH 6.3. In the present study, we extend these results in two regards: (i) The reaction was studied at a series of intermediate pH values under otherwise identical conditions. The new experiments suggest that the transition from perioddoubling to periodadding takes place at approximately pH 5.4 and is characterized by a loss of dynamical complexity. (ii) A detailed model (Bronnikova, T. V.; Fed'kina, V. R.; Schaffer, W. M.; Olsen, L. F. J. Phys. Chem. 1995, 99, 93099312) was shown capable of reproducing the principal experimental results. These include the shift from perioddoubling to periodadding, the existence of narrow regions of perioddoubling between adjacent mixedmode oscillations in the periodadding regime and changes in cycle amplitude.
 Bronnikova, T. V., Schaffer, W. M., & Olsen, L. F. (1996). Quasiperiodicity in a detailed model of the peroxidaseoxidase reaction. Journal of Chemical Physics, 105(24), 1084910859.More infoAbstract: Quasiperiodicity in models of the peroxidaseoxidase reaction has previously been reported in "abstract" or phenomenological models which sacrifice chemical realism for tractability. In the present paper, we discuss how such behavior can arise in a detailed model (BFSO) of the reaction which has previously been shown to be consistent with experimental findings. We distinguish two types of quasiperiodic behavior. Regions of what we here refer to as "primary" quasiperiodicity are delimited by supercritical secondary Hopf bifurcations at one end of the relevant range of parameter values and by heteroclinic transitions at the other. Regions of socalled "secondary quasiperiodicity" are delimited by supercritical Hopf bifurcations at both ends of the parameter range. The existence of a quasiperiodic route to chaos in a modified version of BFSO is also described. This paper emphasizes the experimental circumstances under which quasiperiodic dynamics may be detected in the lab and offers specific prescriptions for its observation. © 1996 American Institute of Physics.
 King, A. A., Schaffer, W. M., Gordon, C., Treat, J., & Kot, M. (1996). Weakly dissipative predatorprey systems. Bulletin of Mathematical Biology, 58(5), 835859.More infoAbstract: In the presence of seasonal forcing, predatorprey models with quadratic interaction terms and weak dissipation can exhibit infinite numbers of coexisting periodic attractors corresponding to cycles of different magnitude and frequency. These motions are best understood with reference to the conservative case, for which the degree of dissipation is, by definition, zero. Here one observes the familiar mix of 'regular' (neutrally stable orbits and tori) and chaotic motion typical of nonintegrable Hamiltonian systems. Perturbing away from the conservative limit, the chaos becomes transitory. In addition, the invariant tori are destroyed and the neutrally stable periodic orbits becomes stable limit cycles, the basins of attraction of which are intertwined in a complicated fashion. As a result, stochastic perturbations can bounce the system from one basin to another with consequent changes in system behavior. Biologically, weak dissipation corresponds to the case in which predators are able to regulate the density of their prey well below carrying capacity.
 Bronnikova, T. V., Fed'kina, V., Schaffer, W. M., & Olsen, L. F. (1995). Perioddoubling bifurcations and chaos in a detailed model of the peroxidaseoxidase reaction. Journal of Physical Chemistry, 99(23), 93099312.More infoAbstract: We propose a detailed model of the peroxidaseoxidase reaction, which is capable of reproducing many of the nonlinear behaviors observed in the experimental system. These include simple and complex periodic oscillations, period doubling bifurcations, chaotic dynamics, and transient chaos. © 1995 American Chemical Society.
 Allen, J. C., Schaffer, W. M., & Rosko, D. (1993). Chaos reduces species extinction by amplifying local population noise. Nature, 364(6434), 229232.More infoPMID: 8321317;Abstract: IN the mid1970s, theoretical ecologists were responsible for stimulating interest in nonlinear dynamics and chaos13. Ironically, the importance of chaos in ecology itself remains controversial417. Proponents of ecological chaos point to its ubiquity in mathematical models and to various empirical findings15,16,18. Sceptics12,19,20 maintain that the models are unrealistic and that the experimental evidence is equally consistent with stochastic models. More generally, it has been argued9,11,2123 that interdemic selection and/or enhanced rates of species extinction will eliminate populations and species that evolve into chaotic regions of parameter space. Fundamental to this opinion is the belief24,25 that violent oscillations and low minimum population densities are inevitable correlates of the chaotic state. In fact, rarity is not a necessary consequence of complex dynamical behaviour26,27. But even when chaos is associated with frequent rarity, its consequences to survival are necessarily deleterious only in the case of species composed of a single population. Of course, the majority of real world species (for example, most insects) consist of multiple populations weakly coupled by migration, and in this circumstance chaos can actually reduce the probability of extinction. Here we show that although low densities lead to more frequent extinction at the local level28, the decorrelating effect of chaotic oscillations reduces the degree of synchrony among populations and thus the likelihood that all are simultaneously extinguished.
 Geest, T., Olsen, L. F., Steinmetz, C. G., Larter, R., & Schaffer, W. M. (1993). Nonlinear analyses of periodic and chaotic time series from the peroxidaseoxidase reaction. Journal of Physical Chemistry, 97(32), 84318441.More infoAbstract: Experimental time series of [O2] generated by the peroxidaseoxidase reaction were analyzed using various methods from nonlinear dynamics. The following results were obtained: (1) Aperiodic time series are chaotic as evidenced by (i) the observation of period doubling bifurcations in response to increasing the amount of 2,4dichlorophenol in the reaction mixture, (ii) welldefined next amplitude maps, (iii) positive Lyapunov exponents corresponding to about 0.5 bits per orbital excursion, and (iv) fractal dimensions ranging from 2.6 to 2.8. Prediction profiles obtained by subjecting the data to nonlinear forecasting give further evidence of deterministic chaos. (2) The chaotic data are nonuniform, since none of the abovementioned methods work well when applied to the continuous data. Instead, one has to use a series of maxima. (3) Conclusions 1 and 2 are supported by the fact that similar results obtain for chaotic time series generated by a simple model of the reaction. © 1993 American Chemical Society.
 Kendall, B. E., Schaffer, W., & Tidd, C. (1993). Transient periodicity in chaos. Physics Letters A, 177(1), 1320.More infoAbstract: Chaotic time series can exhibit rare bursts of "periodic" motion. We discuss one mechanism for this phenomenon of "transient periodicity": the trajectory gets temporarily stuck in the neighborhood of a semiperiodic "semiattractor" (or "chaotic saddle"). This can provide insight for interpreting such phenomena in empirical time series; it also allows for a novel partition of the phase space, in which the attractor may be viewed as the union of many such chaotic saddles. © 1993.
 Schaffer, W. M., Kendall, B. E., Tidd, C. W., & Olsen, L. F. (1993). Transient periodicity and episodic predictability in biological dynamics. IMA J.MATH.APPL.MED.BIOL., 10(4), 227247.More infoPMID: 8133129;
 Schaffer, W. M., Kendall, B. E., Tidd, C. W., & Olsen, L. F. (1993). Transient periodicity and episodic predictability in biological dynamics. Mathematical Medicine and Biology, 10(4), 227247.More infoAbstract: Biological time series often contain passing episodes of nearly periodic dynamics. In chaotic systems, such transient periodicity can reflect the existence of semiperiodic saddlesnonstable invariant setscontained in the attractor. Motion in the vicinity of such objects has a prominent periodic component. In addition, trajectories can become temporarily trapped in these neighbourhoods before exiting. The immediate dynamical antecedents (loworder preimages) of transient periodicity are well defined and, along with the saddles to which they map, correspond to regions of enhanced predictability under nonlinear forecasting. This suggests that it may be possible to forecast the onset of transient periodicity in systems for which overall predictability is low. The present paper reviews these concepts and applies them to biological phenomena at different levels of organization. © 1993 Oxford University Press.
 Tidd, C. W., Olsen, L. F., & Schaffer, W. M. (1993). The case for chaos in childhood epidemics. II. Predicting historical epidemics from mathematical models. Proceedings of the Royal Society B: Biological Sciences, 254(1341), 257273.More infoPMID: 8108458;Abstract: The case for chaos in childhood epidemics rests on two observations. The first is that historical epidemics show various 'fieldmarks' of chaos, such as positive Lyapunov exponents. Second, phase portraits reconstructed from realworld epidemiological time series bear a striking resemblance to chaotic solutions obtained from certain epidemiological models. Both lines of evidence are subject to dispute: the algorithms used to look for the fieldmarks can be fooled by short, noisy time series, and the same fieldmarks can be generated by stochastic models in which there is demonstrably no chaos at all. In the present paper, we compare the predictive abilities of stochastic models with those of mechanistic scenarios that admit to chaotic solutions. The main results are as follows: (i) the mechanistic models outperform their stochastic counterparts; (ii) forecasting efficacy of the deterministic models is maximized by positing parameter values that induce chaotic behaviour; (iii) simple mechanistic models are equal if not superior to more detailed schemes that include age structure; and (iv) prediction accuracy for monthly notifications declines rapidly with time, so that, from a practical standpoint, the results are of little value. By way of contrast, next amplitude maps can successfully forecast successive changes in maximum incidence one or more years into the future.
 Krukonis, G., & Schaffer, W. M. (1991). Population cycles in mammals and birds: Does periodicity scale with body size?. Journal of Theoretical Biology, 148(4), 469493.More infoAbstract: We consider Calder's hypothesis that cycle periods of fluctuating populations of mammalian herbivore scale with the 4th root of body mass. Data adduced by Peterson et al., in support of this hypothesis are reexamined using techniques from spectral analysis. New herbivore data and population statistics for carnivores are also considered. The following results are obtained:o(1)support for the hypothesis does not depend on the method used to assign a period, provided that one is willing to accept cycle periods which are not statistically significant.(2)The explanatory power of the proposed scaling law depends critically on whether or not populations are treated individually or averaged by species.(3)Adding new herbivore species to the Peterson data set decreases the fit and changes the scaling exponent.(4)As originally predicted by Calder, there is no relationship between body mass and cycle period in carnivores./lt. We suggest that Calder's hypothesis represents an expected lower bound when the period is plotted against body mass in the presence of noise and observational errors. © 1991 Academic Press Limited.
 Olsen, L. F., & Schaffer, W. M. (1990). Chaos versus noisy periodicity: Alternative hypotheses for childhood epidemics. Science, 249(4968), 499504.More infoPMID: 2382131;Abstract: Whereas case rates for some childhood diseases (chickenpox) often vary according to an almost regular annual cycle, the incidence of more efficiently transmitted infections such as measles is more variable. Three hypotheses have been proposed to account for such fluctuations, (i) Irregular dynamics result from random shocks to systems with stable equilibria, (ii) The intrinsic dynamics correspond to biennial cycles that are subject to stochastic forcing, (iii) Aperiodic fluctuations are intrinsic to the epidemiology. Comparison of real world data and epidemiological models suggests that measles epidemics are inherently chaotic. Conversely, the extent to which chickenpox outbreaks approximate a yearly cycle depends inversely on the population size.
 Kot, M., Schaffer, W. M., Truty, G. L., Graser, D. J., & Olsen, L. F. (1988). Changing criteria for imposing order. Ecological Modelling, 43(12), 75110.More infoAbstract: Theory is most powerful when it provides new approaches that help resolve longstanding problems. We review the origins, the early development, and some recent trends in the study of chaos and show how new methods of nonlinear dynamics may be used to infer the underlying order in 'noisy' ecological and epidemiological time series. We consider, at some length, the application of these techniques to the study of childhood disease epidemics. © 1988.
 Olsen, L. F., Truty, G. L., & Schaffer, W. M. (1988). Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark. Theoretical Population Biology, 33(3), 344370.More infoPMID: 3266037;Abstract: Using traditional spectral analysis and recently developed nonlinear methods, we analyze the incidence of six childhood diseases in Copenhagen, Denmark. In three cases, measles, mumps, rubella, the dynamics suggest low dimensional chaos. Outbreaks of chicken pox, on the other hand, conform to an annual cycle with noise superimposed. The remaining diseases, pertussis and scarlet fever, remain problematic. The real epidemics are compared with the output of a Monte Carlo analog of the SEIR model for childhood infections. For measles, mumps, rubella, and chicken pox, we find substantial agreement between the model simulations and the data. © 1988.
 Kot, M., & Schaffer, W. M. (1986). Discretetime growthdispersal models. Mathematical Biosciences, 80(1), 109136.More infoAbstract: Integrodifference equations are discretetime models that share many of the attributes of scalar reactiondiffusion equations. At the same time, they readily exhibit period doubling and chaos. We examine the properties of some simple integrodifference equations. © 1986.
 Schaffer, W. M., & Kot, M. (1986). Chaos in ecological systems: The coals that Newcastle forgot. Trends in Ecology and Evolution, 1(3), 5863.More infoAbstract: Until recently, ecologists have ignored the possibility that chaos may be an important component of ecological systems. With the development of powerful new concepts and techniques for the study and detection of chaotic systems, it is becoming apparent that chaos may be widespread in nature, and may necessitate a fundamental reappraisal of many ideas in community and population ecology. © 1986.
 Schaffer, W. M., Ellner, S., & Kot, M. (1986). Effects of noise on some dynamical models in ecology. Journal of Mathematical Biology, 24(5), 479523.More infoPMID: 3805908;Abstract: We investigate effects of random perturbations on the dynamics of onedimensional maps (single species difference equations) and of finite dimensional flows (differential equations for n species). In particular, we study the effects of noise on the invariant measure, on the "correlation" dimension of the attractor, and on the possibility of detecting the nonlinear deterministic component by applying reconstruction techniques to the time series of population abundances. We conclude that adding noise to maps with a stable fixedpoint obscures the underlying determinism. This turns out not to be the case for systems exhibiting complex periodic or chaotic motion, whose essential properties are more robust. In some cases, adding noise reveals deterministic structure which otherwise could not be observed. Simulations suggest that similar results hold for flows whose attractor is almost twodimensional. © 1986 SpringerVerlag.
 Schaffer, W. M. (1985). Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology?. IMA Journal of Mathematics Applied in Medicine and Biology, 2(4), 221252.More infoPMID: 3870986;Abstract: Using recently developed methods in nonlinear dynamics, two hypotheses often advanced to account for recurrent outbreaks of childhood diseases such as measles are investigated. The first, maintenance of otherwise damped oscillations by noise, appears incapable of reproducing essential features of the data. The second, cycles and chaos sustained by seasonal variation in contact rates gives qualitative and quantitative agreement between model and observation. It is concluded that nonlinear dynamics offers a methodology which may allow students of ecology and epidemiology to distinguish between competing mechanistic hypotheses.
 Schaffer, W. M. (1985). Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology?. Mathematical Medicine and Biology, 2(4), 221252.
 Schaffer, W. M. (1985). Order and chaos in ecological systems.. Ecology, 66(1), 93106.More infoAbstract: Systems of differential equations exhibiting complex periodic or chaotic behavior can sometimes be associated with onedimensional mappings (difference equations) which encapsulate the properties of the attractor governing the full ndimensional system. Recent advances make it possible to reconstruct the underlying attractor, and hence the onedimensional map, from timeseries data for a single state variable (species). This technique is illustrated with reference to a hypothetical system consisting of a single predator species and 2 species of victims. Trapping records for Canadian lynx Lynx canadensis suggest that the method may have application to realworld populations. Author
 Schaffer, W. M., & Kot, M. (1985). Nearly one dimensional dynamics in an epidemic. Journal of Theoretical Biology, 112(2), 403427.More infoPMID: 3982045;Abstract: The incidence of measles in New York City and Baltimore was studied using recently developed techniques in nonlinear dynamics. The data, monthly case reports for the years 19281963, suggest almost two dimensional, chaotic flows whose essential attributes are captured by one dimensional, unimodal maps. The effects of noise, inevitable in ecological and epidemiological systems are discussed. © 1985 Academic Press Inc. (London) Ltd.
 Kot, M., & Schaffer, W. M. (1984). The effects of seasonality on discrete models of population growth. Theoretical Population Biology, 26(3), 340360.More infoAbstract: The effects of seasonality on the dynamics of a bivoltine population with discrete, nonoverlapping generations are examined. It is found that large seasonality is inevitably destabilizing but that mild seasonality may have a pronounced stabilizing effect. Seasonality also allows for the coexistence of alternative stable states (equilibria, cycles, chaos). These solutions may be seasonally inphase, outofphase, or asynchronous. Inphase solutions correspond to winter regulation of population density, whereas outofphase solutions correspond to summer regulation. Analysis suggests that summer regulation is possible only in mildly seasonal habitats. © 1984.
 Schaffer, W. M. (1983). The application of optimal control theory to the general life history problem.. American Naturalist, 121(3), 418431.More infoAbstract: The application of optimal control theory to life history evolution in species with discrete breeding seasons and overlapping generations is discussed. For each age class, the objective functional maximized consists of an integral (total reproduction for that age class) plus a final function (residual reproductive value) . A simple example, for which monocarpy is the optimal strategy, is given. from Author
 Schaffer, W. M., Zeh, D. W., Buchmann, S. L., Kleinhans, S., Schaffer, M. V., & Antrim, J. (1983). Competition for nectar between introduced honey bees and native North American bees and ants.. Ecology, 64(3), 564577.More infoAbstract: Previous studies of introduced honey bees foraging at Agave schottii flowers suggest that Apis mellifera preferentially exploits the most productive patches of flowers and thereby reduces the standing crop of available nectar and the utilization of these sites by native bees. Results of experiments undertaken to evaluate this hypothesis are given and discussed using Apis, Bombus and Xylocopa. Authors
 Schaffer, W. M., Inouye, R. S., & Whittam, T. S. (1982). Energy allocation by an annual plant when the effects of seasonality on growth and reproduction are decoupled.. American Naturalist, 120(6), 787815.More infoAbstract: A model of an annual plant is developed in which it is assumed that photosynthate can be allocated to new leaves or seeds, or stored for subsequent allocation to reproduction. Photosynthetic rate and reproductive efficiency are allowed to vary seasonally with the period of maximum photosynthesis preceding the date of maximum reproductive efficiency. Using methods from optimal control theory, the model is solved and the circumstances favoring the storage of materials determined.Authors
 Schaffer, W. M. (1979). Equivalence of maximizing reproductive value and fitness in the case of reproductive strategies. Proceedings of the National Academy of Sciences of the United States of America, 76(7), 35673569.More infoPMID: 16592685;PMCID: PMC383869;Abstract: The proposed equivalence of maximizing reproductive value and fitness is examined for two model life histories. In the first instance, it is assumed that offspring are fledged before the start of the next breeding season. In this case the proposed equivalence is verified. In the second model, parents care for their progeny for more than 1 year. In this case the optimal reproductive expenditure at a particular age is shown to depend on both current reproductive value and the diminution in survival rates of previously conceived young still dependent on parental protection.
 Rosenzweig, M. L., & Schaffer, W. M. (1978). Homage to the red queen. II. Coevolutionary response to enrichment of exploitation ecosystems. Theoretical Population Biology, 14(1), 158163.More infoPMID: 741395;Abstract: The model of predatorvictim coevolution developed in the preceding paper (Schaffer and Rosenzweig, 1978) is used to analyze the coevolutionary response to ecosystem enrichment. It is shown that coevolution tends to oppose the destabilization that results from the enrichment itself. In the circumstance that the victims' r and K are proportional, gradual enrichment followed by coevolutionary adjustment actually enhances ecosystem stability. © 1978.
 Schaffer, W. M., & Rosenzweig, M. L. (1978). Homage to the red queen. I. Coevolution of predators and their victims. Theoretical Population Biology, 14(1), 135157.More infoPMID: 741394;Abstract: In this paper we develop a model to describe the coevolution of a twospecies predatorvictim ecosystem. Evolution among the predators acts to displace the predators' zero growth isocline, J, to the left, whereas victim evolution has the opposite effect. More important, we show that for many such systems, there is a value of J, J*, at which the rate of predator evolution exactly equals the rate of victim evolution. This value of J turns out to be a coevolutionary steady state (CSS) which is stable to perturbations. It is suggested that this analysis permits qualitative understanding of why certain real world predatorprey systems oscillate, whereas others do not. © 1978.
 Schaffer, W. M. (1977). Evolution, population dynamics, and stability: A comment. Theoretical Population Biology, 11(3), 326329.More infoPMID: 877910;Abstract: A recently proposed criterion for community coevolution is shown to be equivalent to requiring that a mutant have a positive selective advantage if it is to spread through a population. This is one of the basic conclusions of deterministic population genetics. © 1977.
 Schaffer, W. M. (1977). Some observations on the evolution of reproductive rate and competitive ability in flowering plants. Theoretical Population Biology, 11(1), 90104.More infoPMID: 854863;
 Schaffer, W. M. (1965). Charles Lyell and the origin of species. Yale Scientfic Magazine, 39(8), 1014.