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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
19 Oct 2001-Science
TL;DR: This work compared the predictions of discrete-state and continuous-state population models and suggested that such lattice effects could be an important component of natural population fluctuations.
Abstract: Animals and many plants are counted in discrete units. The collection of possible values (state space) of population numbers is thus a nonnegative integer lattice. Despite this fact, many mathematical population models assume a continuum of system states. The complex dynamics, such as chaos, often displayed by such continuous-state models have stimulated much ecological research; yet discrete-state models with bounded population size can display only cyclic behavior. Motivated by data from a population experiment, we compared the predictions of discrete-state and continuous-state population models. Neither the discrete- nor continuous-state models completely account for the data. Rather, the observed dynamics are explained by a stochastic blending of the chaotic dynamics predicted by the continuous-state model and the cyclic dynamics predicted by the discrete-state models. We suggest that such lattice effects could be an important component of natural population fluctuations.

88 citations

Journal ArticleDOI
01 Jun 2008-Ecology
TL;DR: The erratic pattern of growth of the population of reindeer in Adventdalen seems to result from a combination of the effects of nonlinear density dependence, strong density-dependent mortality, and variable density independence related to ablation in winter.
Abstract: Nonlinear and irregular population dynamics may arise as a result of phase dependence and coexistence of multiple attractors. Here we explore effects of climate and density in the dynamics of a highly fluctuating population of wild reindeer (Rangifer tarandus platyrhynchus) on Svalbard observed over a period of 29 years. Time series analyses revealed that density dependence and the effects of local climate (measured as the degree of ablation [melting] of snow during winter) on numbers were both highly nonlinear: direct negative density dependence was found when the population was growing (Rt > 0) and during phases of the North Atlantic Oscillation (NAO) characterized by winters with generally high (1979-1995) and low (1996-2007) indices, respectively. A growth-phase-dependent model explained the dynamics of the population best and revealed the influence of density-independent processes on numbers that a linear autoregressive model missed altogether. In particular, the abundance of reindeer was enhanced by ablation during phases of growth (Rt > 0), an observation that contrasts with the view that periods of mild weather in winter are normally deleterious for reindeer owing to icing of the snowpack. Analyses of vital rates corroborated the nonlinearity described in the population time series and showed that both starvation mortality in winter and fecundity were nonlinearly related to fluctuations in density and the level of ablation. The erratic pattern of growth of the population of reindeer in Adventdalen seems, therefore, to result from a combination of the effects of nonlinear density dependence, strong density-dependent mortality, and variable density independence related to ablation in winter.

81 citations

Journal ArticleDOI
TL;DR: It is suggested that it may be challenging to forecast recruitment phase transitions, though it is encouraged to determine whether there are unifying relationships that govern recruitment dynamics, and to propose that transitions in recruitment are dependent upon the balance of activating and constraining processes of recruitment control.
Abstract: There has been a recent resurgence of interest in fisheries recruitment as a dynamic and complex process that is integrated over several life stages, with a variety of factors acting across scales to initiate, modulate, and constrain population abundance and variability. In this paper, we review the theory of recruitment phase transitions using a marine fisheries perspective. We propose that transitions in recruitment are dependent upon the balance of activating and constraining processes of recruitment control, and we highlight fundamental differences in recruitment transitions precipitated by climate events, those related to community alterations, and those manifested by fishery practices (though each is not necessarily mutually exclusive). We maintain that the emergent properties of fisheries populations post-phase transition are contingent upon their histories, their differing initial states, the degree of food web complexity, interaction strengths among interspecifics, and contrasting external forcing, any or all of which may be dissimilar between one regime and another. We suggest that it may be challenging to forecast recruitment phase transitions, though we encourage efforts to determine whether there are unifying relationships that govern recruitment dynamics.

59 citations

Journal ArticleDOI
TL;DR: In this paper, a mathematical analysis of the critical domain-size problem for structured populations is presented, where space is introduced explicitly into matrix models for stage-structured populations and movement of individuals is described by means of a dispersal kernel.
Abstract: This paper is concerned with mathematical analysis of the 'critical domain-size' problem for structured populations. Space is introduced explicitly into matrix models for stage-structured populations. Movement of individuals is described by means of a dispersal kernel. The mathematical analysis investigates conditions for existence, stability and uniqueness of equilibrium solutions as well as some bifurcation behaviors. These mathematical results are linked to species persistence or extinction in connected habitats of different sizes or fragmented habitats; hence the framework is given for application of such models to ecology. Several approximations which reduce the complexity of integrodifference equations are given. A simple example is worked out to illustrate the analytical results and to compare the behavior of the integrodifference model to that of the approximations.

53 citations


Additional excerpts

  • ...Matrix models have been applied in many different areas like fisheries [7], insects [15] or tree management [28], to name but a few....

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Journal ArticleDOI
TL;DR: A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume.
Abstract: Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable ’saddle cycle’ separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.

48 citations


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

  • ...An exception is the laboratory experiment conducted by Jillson (1980) and the subsequent model-based explanation proposed by Costantinoet al. (1998). For a different interpretation of the Jillson experiment see Nisbet and Gurney (1981 ) and Renshaw (1991 )....

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  • ...An exception is the laboratory experiment conducted by Jillson (1980) and the subsequent model-based explanation proposed by Costantinoet al....

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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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