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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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Book ChapterDOI
01 Jan 2014
TL;DR: A survey of discrete-time mathematical models that address this issue can be found in this paper, where the authors point out the main mathematical methods used to prove the occurrence of competitive exclusion in these models.
Abstract: In biology, the principle of competitive exclusion, largely attributed to the Russian biologist G. F. Gause, states that two species competing for common resources (food, territory etc.) cannot coexist, and that one of the species drives the other to extinction. We make a survey of discrete-time mathematical models that address this issue and point out the main mathematical methods used to prove the occurrence of competitive exclusion in these models. We also offer examples of models in which competitive exclusion fails to take place, or at least it is not the only outcome. Finally, we present an extension of the competitive exclusion results in [1, 5] to a more general model.

3 citations

Book ChapterDOI
01 Jan 2007
TL;DR: Theoretical studies of population dynamics and ecological interactions tend to focus on asymptotic attractors of mathematical models as discussed by the authors, and experimental studies show that even in controlled laboratory conditions the attractors are likely to be insufficient to explain observed temporal patterns in data.
Abstract: Theoretical studies of population dynamics and ecological interactions tend to focus on asymptotic attractors of mathematical models. Modeling and experimental studies show, however, that even in controlled laboratory conditions the attractors of mathematical models are likely to be insufficient to explain observed temporal patterns in data. Instead, one is more likely to see a collage of many patterns that resemble various dynamics predicted by a deterministic model that arise during randomly occurring temporal episodes. These deterministic “signals” might include patterns characteristic of a model attractor (or several model attractors — even from possibly different deterministic models), transients both near and far from attractors, and/or unstable invariant sets and their stable manifolds. This paper discusses several examples taken from experimental projects in population dynamics that illustrate these and other tenets.

1 citations

01 Jan 2015
TL;DR: A brief survey of nonlinear Leslie models focusing on the fundamental bifurcation that occurs when the extinction equilibrium destabilizes as 0 increases through 1 is given in this article.
Abstract: This brief survey of nonlinear Leslie models focuses on the fundamental bifurcation that occurs when the extinction equilibrium destabilizes as 0 increases through 1. Of particular interest is the bifurcation that occurs when only the oldest age class is reproductive, in which case the Leslie projection matrix is not primitive. This case is distinguished by the invariance of the boundary of the positive cone on which orbits contain temporally synchronized, missing age classes and by the bifurcation of oscillatory attractors, lying on the boundary of the positive cone, in addition to the bifurcation of positive equilibria. The lack of primitivity of the Leslie projection matrix, while seemingly only a mathematically technicality, corresponds to a fundamental life history strategy in population dynamics, namely, semelparity (when individuals have one reproductive event before dying). The study of semelparous Leslie models was historically motivated by the synchronized outbreak cycles of periodical insects, the most famous being the long-lived cicadas (C. magicicada spp).
Journal ArticleDOI
TL;DR: In this article , a model of the spatio-temporal dynamics of two-age-structured populations coupled by migration (metapopulation) with long-range displacement is proposed.
Abstract: The inhomogeneous population distribution appears as various population densities or different types of dynamics in distant sites of the extended habitat and may arise due to, for example, the resettlement features, the internal population structure, and the population dynamics synchronization mechanisms between adjacent subpopulations. In this paper, we propose the model of the spatio-temporal dynamics of two-age-structured populations coupled by migration (metapopulation) with long-range displacement. We study mechanisms leading to inhomogeneous spatial distribution as a type of cluster synchronization of population dynamics. To study the spatial patterns and synchronization, we use the method of constructing spatio-temporal profiles and spatial return maps. We found that patterns with spots or stripes are typical spatial structures with synchronous dynamics. In most cases, the spatio-temporal dynamics are mixed with randomly located single populations with strong burst (outbreak) of population size (solitary states). As the coupling parameters decrease, the number of solitary states grows, and they increasingly synchronize and form the clusters of solitary states. As a result, there are the several clusters with different dynamics. The appearance of these spatial patterns most likely occurs due to the multistability of the local age-structured population, leading to the spatio-temporal multistability.
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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