Phase switching in population cycles

doi:10.1098/RSPB.1998.0564

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Phase switching in population cycles

Journal Article•DOI•

Phase switching in population cycles

Shandelle M. Henson, Jim M. Cushing¹, R. F. Costantino², Brian Dennis³, Robert A. Desharnais⁴ - Show less +1 more•Institutions (4)

University of Arizona¹, University of Rhode Island², University of Idaho³, California State University, Los Angeles⁴

22 Nov 1998-Proceedings of The Royal Society B: Biological Sciences (The Royal Society)-Vol. 265, Iss: 1411, pp 2229-2234

TL;DR: In this paper, 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, which accounts for two-cycle phase shifts and the occurrence of asynchronous replicates in experimental cultures of Tribolium.

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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 .

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Journal Article•DOI•

Park's Tribolium competition experiments: a non-equilibrium species coexistence hypothesis

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Jeffrey Edmunds¹, Jim M. Cushing², R. F. Costantino², Shandelle M. Henson³, Brian Dennis⁴, Robert A. Desharnais⁵ - Show less +2 more•Institutions (5)

University of Mary Washington¹, University of Arizona², Andrews University³, University of Idaho⁴, California State University, Los Angeles⁵

01 Sep 2003-Journal of Animal Ecology

TL;DR: It is suggested 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, in contradiction to classical tenets of competition theory.

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Abstract: Summary 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.

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47 citations

Cites background from "Phase switching in population cycle..."

...…of Arizona, Tucson, AZ, 85721, § Department of Mathematics, Andrews University, Berrien Springs, MI 49104; ¶ epartment of Fish and Wildlife Resources and Division of Statistics, University of Idaho, Moscow, ID, 83844; and ** Department of Biological Sciences, California State University, Los...
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...…et al. 1995, 1997, 2001; Costantino et al. 1997), multiple attractors and resonance (Costantino et al. 1998; Henson et al. 2002), phase switching (Henson et al. 1998), saddle influences (Cushing et al. 1998), the use of small perturbations to control insect outbreaks (Desharnais et al. 2001),…...
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Journal Article•DOI•

Long-term shifts in the cyclicity of outbreaks of a forest-defoliating insect

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Andrew J. Allstadt¹, Kyle J. Haynes¹, Andrew M. Liebhold², Derek M. Johnson³•Institutions (3)

University of Virginia¹, United States Forest Service², Virginia Commonwealth University³

01 May 2013-Oecologia

TL;DR: Wavelet analysis to an 86-year time series of forest defoliation in the northeastern United States suggests that changes in gypsy moth population behavior are driven by trophic interactions, rather than by changes in climatic conditions frequently implicated in other systems.

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Abstract: Recent collapses of population cycles in several species highlight the mutable nature of population behavior as well as the potential role of human-induced environmental change in causing population dynamics to shift. We investigate changes in the cyclicity of gypsy moth (Lymantria dispar) outbreaks by applying wavelet analysis to an 86-year time series of forest defoliation in the northeastern United States. Gypsy moth population dynamics shifted on at least four occasions during the study period (1924–2009); strongly cyclical outbreaks were observed between ca. 1943–1965 and ca. 1978–1996, with noncyclical dynamics in the intervening years. During intervals of cyclical dynamics, harmonic oscillations at cycle lengths of 4–5 and 8–10 years co-occurred. Cross-correlation analyses indicated that the intensity of suppression efforts (area treated by insecticide application) did not significantly reduce the total area of defoliation across the region in subsequent years, and no relationship was found between insecticide use and the cyclicity of outbreaks. A gypsy moth population model incorporating empirically based trophic interactions produced shifting population dynamics similar to that observed in the defoliation data. Gypsy moth cycles were the result of a high-density limit cycle driven by a specialist pathogen. Though a generalist predator did not produce an alternative stable equilibrium, cyclical fluctuations in predator density did generate extended intervals of noncyclical behavior in the gypsy moth population. These results suggest that changes in gypsy moth population behavior are driven by trophic interactions, rather than by changes in climatic conditions frequently implicated in other systems.

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46 citations

Cites background from "Phase switching in population cycle..."

...In these cases, perturbations to the system (e.g., stochastic effects) can cause a population’s trajectory to jump from one attractor to another (Henson et al. 1998; Bauch and Earn 2003; Dwyer et al. 2004)....
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...Empirical support for these theoretical predictions stems largely from studies done on laboratory populations of flour beetles (Tribolium) (Costantino et al. 1995; Dennis et al. 1997; Henson et al. 1998)....
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..., stochastic effects) can cause a population’s trajectory to jump from one attractor to another (Henson et al. 1998; Bauch and Earn 2003; Dwyer et al. 2004)....
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...Henson et al. (1998), for example, showed that shifts in the cyclicity of Tribolium beetles resulted from stochastic jumps between basins of attraction....
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Book Chapter•DOI•

Nonlinear Stochastic Population Dynamics: The Flour Beetle Tribolium as an Effective Tool of Discovery

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R. F. Costantino, Robert A. Desharnais, Jim M. Cushing, Brian Dennis, Shandelle M. Henson, Aaron A. King - Show less +2 more

01 Jan 2005-Advances in Ecological Research

TL;DR: The best procedure, as in the rest of science, is first to simplify the system, then to hold it more or less constant while varying the important parameters one or two at a time to see what happens.

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Abstract: When observation and theory collide, scientists turn to carefully designed experiments for resolution. Their motivation is especially high in the case of biological systems, which are typically far too complex to be grasped by observation and theory alone. The best procedure, as in the rest of science, is first to simplify the system, then to hold it more or less constant while varying the important parameters one or two at a time to see what happens. —Edward O. Wilson (2002)

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42 citations

Cites background from "Phase switching in population cycle..."

...We hypothesize 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 (Henson et al. 1998)....
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...We hypothesize 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 (Henson et al. 1998)....
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Journal Article•DOI•

A chaotic attractor in ecology: theory and experimental data

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Jim M. Cushing¹, Shandelle M. Henson², Robert A. Desharnais³, Brian Dennis⁴, R. F. Costantino⁵, Aaron A. King¹ - Show less +2 more•Institutions (5)

University of Arizona¹, College of William & Mary², California State University, Los Angeles³, University of Idaho⁴, University of Rhode Island⁵

02 Jan 2001-Chaos Solitons & Fractals

TL;DR: In this article, a deterministic LPA model was used to predict the behavior of a population of flour beetles (Tribolium castaneum) in controlled laboratory experiments, including a specific route to chaos.

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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.

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39 citations

Cites background or methods from "Phase switching in population cycle..."

...We have used the LPA model to provide previously unavailable explanations for observed patterns in data or suggested patterns previously unobserved, including temporal and phase space patterns caused by stable manifolds of unstable saddles [6,12], phase switching in oscillating populations [13], and unexpected resonances due to periodic habitats [15,16]....
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..., the in ̄uence of unstable invariant sets) [12,13], multiple attractors and their basins of attraction [13,14], resonances in periodically forced habitats [15,16], and many others....
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Journal Article•DOI•

Basins of attraction: population dynamics with two stable 4‐cycles

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Shandelle M. Henson¹, R. F. Costantino², Robert A. Desharnais³, Jim M. Cushing, Brian Dennis⁴ - Show less +1 more•Institutions (4)

Andrews University¹, University of Rhode Island², California State University, Los Angeles³, University of Idaho⁴

01 Jul 2002-Oikos

TL;DR: In this article, the authors use composite maps, basins of attraction, basin switching, and saddle fly-by's to make the existence of multiple attractors more accessible to experimental scrutiny.

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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.

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27 citations

Cites background or methods from "Phase switching in population cycle..."

...However, if the model is composed with itself three times, so that orbits correspond to every fourth step of the period-4 model, we obtain an autonomous composite map whose phase portrait facilitates visualization of both model predictions and the data (Henson et al. 1998, 1999)....
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...Basin switching and the influence of saddles has been documented, by means of the LPA model, in several different Tribolium experiments (Cushing et al. 1998a, b, Henson et al. 1998, 1999)....
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...In some cases, a stochastic event can place a culture directly into another attractor basin (Henson et al. 1998)....
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References

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Book•

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

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John Guckenheimer, Philip Holmes, M. Slemrod¹•Institutions (1)

Rensselaer Polytechnic Institute¹

01 Aug 1983

TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.

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Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.

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12,669 citations

A Reflection on Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

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J. Guckenheimer, P. J. Holmes

01 Jan 2015

TL;DR: In this paper, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.

...read moreread less

12,485 citations

Book•

The struggle for existence

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G. F. Gauze

01 Jan 1934

TL;DR: For three-quarters of a century past more has been written about natural selection and the struggle for existence that underlies the selective process, than perhaps about any other single idea in the whole realm of Biology as discussed by the authors.

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Abstract: For three-quarters of a century past more has been written about natural selection and the struggle for existence that underlies the selective process, than perhaps about any other single idea in the whole realm of Biology. We have seen natural selection laid on its Sterbebett, and subsequently revived again in the most recent times to a remarkable degree of vigor. There can be no doubt that the old idea has great survival value.

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2,641 citations

Journal Article•DOI•

Bifurcations and Dynamic Complexity in Simple Ecological Models

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Robert M. May, George Oster

01 Jul 1976-The American Naturalist

TL;DR: It is shown that as a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes".

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Abstract: Many biological populations breed seasonally and have nonoverlapping generations, so that their dynamics are described by first-order difference equations, Nt+1 = F (Nt). In many cases, F(N) as a function of N will have a hump. We show, very generally, that as such a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes." We give a detailed account of the underlying mathematics of this process and review other situations (in two- and higher dimensional systems, or in differential equation systems) where apparently random dynamics can arise from bifurcation processes. This complicated behavior, in simple deterministic models, can have disturbing implications for the analysis and interpretation of biological data.

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1,119 citations

"Phase switching in population cycle..." refers methods in this paper

...The idea is to look at the population at every other time-step by constructing the so-called c̀omposite' map (e.g. see May & Oster 1976)....
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Book•

An introduction to structured population dynamics

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Jim M. Cushing

01 Jan 1987

TL;DR: In this paper, the authors present a case study of multispecies interactions with continuous models of age-structured models and show that these models can be used in a variety of applications.

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Abstract: Preface 1 Discrete Models Matrix Models Autonomous Single Species Models Some Applications A Case Study Multispecies Interactions 2 Continuous Models Age-Structured Models Autonomous Age-Structured Models Some Applications Multispecies Interactions Other Structured Models 3 Population Level Dynamics Ergodicity and Nonlinear Models The Linear Chain Trick Hierarchical Models Total Population Size in Age-Structured Models Appendix A Stability Theory for Maps Linear Maps Linearization of Maps Appendix B Bifurcation Theorems A Global Bifurcation Theorem Local Parameterization Appendix C Miscellaneous Proofs Bibliography Index

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599 citations

"Phase switching in population cycle..." refers background or methods in this paper

...…including the transitions between dynamic regimes (such as equilibria, two-cycles, three-cycles, invariant loops, and chaos), multiple attractors, and saddle in£uences (Costantino et al. 1995, 1997, 1998; Cushing 1996, 1998; Dennis et al. 1995, 1997; Desharnais et al. 1997; Henson et al. 1999)....
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...Local stability results for both the LPA model and its composite are obtained using standard linearization techniques (Cushing 1998; Guckenheimer & Holmes 1983)....
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...Local stability results for both the LPA model and its composite are obtained using standard linearization techniques (Cushing1998; Guckenheimer & Holmes1983)....
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...it is a saddle point; see Cushing et al. (1998))....
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