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Journal ArticleDOI

Phase switching in population cycles

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.
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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Citations
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Journal ArticleDOI
TL;DR: It is found that, for intermediate values of the dominance of a pest gene for resistance to the toxin, the local refuge can spoil the benefit that is provided by the immigrant stream.

11 citations

Journal ArticleDOI
TL;DR: A discrete stage-structured model of two competing species is derived from a well-tested single-species model of insect populations, and is shown to exhibit multiple attractors for parameter values similar to those used in laboratory experiments which demonstrated chaos in population dynamics.

10 citations

Journal ArticleDOI
TL;DR: This work combines chemostat experiments with dynamical modeling to study the response of the phytoplankton species Chlorella vulgaris to press perturbations, and shows how identical future steady states can be approached via different transients depending on the initial population structure.
Abstract: Stage structures of populations can have a profound influence on their dynamics. However, not much is known about the transient dynamics that follow a disturbance in such systems. Here we combined chemostat experiments with dynamical modeling to study the response of the phytoplankton species Chlorella vulgaris to press perturbations. From an initially stable steady state, we altered either the concentration or dilution rate of a growth-limiting resource. This disturbance induced a complex transient response—characterized by the possible onset of oscillations—before population numbers relaxed to a new steady state. Thus, cell numbers could initially change in the opposite direction of the long-term change. We present quantitative indexes to characterize the transients and to show that the dynamic response is dependent on the degree of synchronization among life stages, which itself depends on the state of the population before perturbation. That is, we show how identical future steady states can b...

7 citations

Journal ArticleDOI
TL;DR: In this paper, the authors use concepts from dynamical systems theory to present a model-based method for quantifying the risk of impending cycle irregularity, such as episodes of damped oscillation and abrupt changes of cycle phase.
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.

5 citations


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

  • ...Noise, always present in population dynamics, causes cycle irregularities such as outbreaks, switches in oscillation phase [28], and episodes of damped oscillations caused by “saddle fly-bys” [11] or other stochastic visitations of unstable equilibria....

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  • ...The discrete stage-structured “LPA” Tribolium model has successfully explained and predicted nonlinear phenomena in a variety of contexts, including transitions between dynamic regimes (such as equilibria, 2-cycles, 3-cycles, invariant loops, and chaos), multiple attractors, saddle influences, stable and unstable manifolds, and lattice effects [4, 7, 5, 10, 8, 11, 12, 9, 13, 14, 15, 17, 18, 28, 24, 27, 25, 26, 30]....

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Journal ArticleDOI
TL;DR: In this article, the authors present developed approaches and mathematical discrete-time models to study the emergence of multistability, synchronization, and clustering in population systems, and the second part of this article focuses on subjects devoted to mathematical modeling of the evolution of limited populations and migration affecting the dynamics of coupled populations and the patterns of their spatial distribution.
Abstract: The second part of this article focuses on subjects devoted to mathematical modeling of the evolution of limited populations and migration affecting the dynamics of coupled populations and the patterns of their spatial distribution. The purpose of this article is to present developed approaches and mathematical discrete-time models to study the emergence of multistability, synchronization, and clustering in population systems.

4 citations

References
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Book
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.
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.

12,669 citations

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.
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.

12,485 citations

Book
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.
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.

2,641 citations

Journal ArticleDOI
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".
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.

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
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.
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

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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