Phase switching in population cycles
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.
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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TL;DR: It is shown that predictions based on the deterministic fixed‐development approach are differentially sensitive to variability and noise in key life stages, whereas generation cycles are robust to the intrinsic variability and uncertainty that may be found in nature.
Abstract: Few age‐structured models of species dynamics incorporate variability and uncertainty in population processes Motivated by laboratory data for an insect and its parasitoid, we investigate whether such assumptions are appropriate when considering the population dynamics of a single species and its interaction with a natural enemy Specifically, we examine the effects of developmental variability and demographic stochasticity on different types of cyclic dynamics predicted by traditional models We show that predictions based on the deterministic fixed‐development approach are differentially sensitive to variability and noise in key life stages In particular, we find that the demonstration of half‐generation cycles in the single‐species model and the multigeneration cycles in the host‐parasitoid model are sensitive to the introduction of developmental variability and noise, whereas generation cycles are robust to the intrinsic variability and uncertainty that may be found in nature
23 citations
Cites background from "Phase switching in population cycle..."
...…body of work shows that incorporating stochasticity into a broad spectrum of models can have important dynamical consequences (Higgins et al. 1997; Henson et al. 1998; Bjørnstad and Grenfell 2001; Coulson et al. 2001; Rohani et al. 2002; Turchin 2003) as well as strongly influence population…...
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TL;DR: There are situations when local populations have different (and sometimes radically different) modes of population dynamics at the same values of demographic parameters, which is referred to as multistability in the theory of dynamic systems.
Abstract: 42 Fluctuation of animal populations continues to be one of the most interesting and mysterious ecological phenomena. A lot of evidence available to date sugg gests not only regular changes in population abunn dance, but also clear cycling of various dynamic modes in biological populations. The most striking and welllknown examples of changes in animal population dynamics are transitions between stable and cyclic phases in the lemming (Lemmus lemmus) populations all over southern Norr way [1] and in the reddgray vole (Clethrionomys rufocaa nus) populations in Finland [2]. The reverse situation can be also observed, when small fluctuations around the equilibrium state are replaced by either oscillatory or chaotic modes. For example, a trend of the snow goose population (Chen caerulescens) in the New York state (United States) to grow up monotonously for a long time has changed to irregular oscillations, probaa bly, because of exceeding the ecological capacity of the environment [3]. Another type of disturbed populaa tion dynamics is related to variation in the lengths of cycles. In particular, in Canada and the northern United States, there was a transition from twoo to threeeyear oscillations in the evening grosbeak (Coccoo thraustes vespertinus) population [4]. Note that similar phenomena were observed in the populations of lemm ming and some vole species, because the cycles with lengths of 2, 3, and 4 are characteristic of them [5, 6]. Moreover, there are situations when nonninteracting, practically identical populations of the same species display different dynamics. In particular, the laboraa tory experiments have demonstrated that, at the same initial population size and under similar conditions, two different antiiphased periodic modes may be observed in the flour beetle (Tribolium castaneum) populations [7]. Thus, there are situations when local populations have different (and sometimes radically different) modes of population dynamics at the same values of demographic parameters. This phenomenon of the dynamic mode dependence on initial conditions is referred to as multistability in the theory of dynamic systems [8]. Appearance of several different dynamic modes is possible when a system has several stable attractors, each serving as a stable point or being involved into a limiting set of attractors (e.g., an invariant curve). Hence, the term \" multistability \" is somewhat misleading in this context and, to our opinn ion, it is more convenient to use a new notion of \" mull timode \" to reflect the essence of the phenomenon occurring in real …
18 citations
01 Oct 2005
TL;DR: In this article, the competitive exclusion principle of the Lotka/Volterra model was used in conjunction with the Leslie/Gower model in the first half of the last century for a competition for species of Tribolium (flour beetles).
Abstract: One of the fundamental tenets of ecology is the Competitive Exclusion Principle According to this principle too much interspecific competition between two species results in the exclusion of one species This Principle is supported by a wide variety of theoretical models, of which the Lotka/Volterra model based on differential equations is the most familiar It is perhaps less well known that difference equations also played an important role in the historical development of the Competitive Exclusion Principle The Leslie/Gower model was used in conjunction with influential competition experiments using species of Tribolium (flour beetles) carried out in the first half of the last century This difference equation model exhibits the same dynamic scenarios as does the Lotka/Volterra model and also supports the Competitive Exclusion Principle A recently developed competition for Tribolium species, however, exhibits a larger variety of dynamic scenarios and competitive outcomes, some of which seemingly stand in contradiction to the Principle We discuss features of this model that differentiate it from the Leslie/Gower model We give a simpler, lower dimensional “toy” model that illustrates some non-Lotka/Volterra dynamics AMS Nos 39A11, 92D40
16 citations
TL;DR: In this paper, a phenomenon of multiregimism is found in the elementary mathematical model of population dynamics, meaning the possibility for different dynamic regimes to exist under the same conditions, with transition to these regimes dependent on the initial numerical values.
Abstract: There is a phenomenon of multiregimism found in the elementary mathematical model of population dynamics, meaning the possibility for different dynamic regimes to exist under the same conditions, with transition to these regimes dependent on the initial numerical values. The effect in question comes into existence in the model which has several different limiting regimes (attractors): equilibrium, regular fluctuations, and chaotic attractor. The revealed phenomenon of multiregimism lets us explain the initiation of fluctuations as well as disappearance of fluctuations. Adequacy of the model's dynamic regimes is depicted by their correlation with the actual dynamics of population size of bank vole ( Myodes glareolus ). It is shown that the impact of climatic factors on a reproductive process of a population noticeably extends the range of possible dynamic regimes and, in fact, leads to random migration over attraction basins of these regimes.
15 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 R 0 increases through 1 is given in this paper.
Abstract: This brief survey of nonlinear Leslie models focuses on the fundamental bifurcation that occurs when the extinction equilibrium destabilizes as R0 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).
15 citations
References
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Book•
01 Aug 1983TL;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
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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