scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Multiple attractors, saddles, and population dynamics in periodic habitats

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.
About: This article is published in Bulletin of Mathematical Biology.The article was published on 1999-10-01. It has received 48 citations till now. The article focuses on the topics: Population & Saddle.
Citations
More filters
Journal ArticleDOI
01 Aug 2003-Oikos
TL;DR: In this article, it was shown that the sensitivity of a stochastic Lyapunov exponent (SLE) to the initial conditions of a deterministic model is not a sign of chaos.
Abstract: An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that “chaos” should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

264 citations


Cites background from "Multiple attractors, saddles, and p..."

  • ...…three-cycles, invariant loops, and chaos), multiple attractors, saddle influences, sensitivity to initial conditions, and lattice effects (Costantino et al. 1995, 1997, 1998, Cushing et al. 1996, 1998a, b, 2001, Dennis et al. 1995, 1997, 2001, Henson et al. 1999, 2001, Desharnais et al. 2001)....

    [...]

Journal ArticleDOI
01 Mar 2008-Ecology
TL;DR: An overview of theoretical themes related to resource pulses is provided, suggesting short-term responses to a single pulse can qualitatively differ from longer-term responds.
Abstract: Over the last several decades, there has been a growing appreciation of the importance of nonequilibrial phenomena and transient dynamics in explaining the structure of ecological communities. This paper provides an overview of theoretical themes related to resource pulses. Theoretical models suggest short-term responses to a single pulse can qualitatively differ from longer-term responses. Recurrent resource pulses can alter community structure, permitting coexistence that otherwise would not occur, or hamper coexistence mechanisms effective in stable environments. For a given resource input, system responses can be more dramatic with short pulses. Resource pulses can cause transitions between alternative states. Dispersal permits species to exploit locally sporadic resource pulses and persist in environments that on average are unsuitable. All these issues are ripe for further theoretical explorations.

246 citations


Cites background from "Multiple attractors, saddles, and p..."

  • ...…extended the Ricker model for single-species density-dependent systems to include periodic fluctuations in resources (Henson and Cushing 1997, Henson et al. 1999, Henson 2000) so as to interpret laboratory experiments with Tribolium beetles sustained by fluctuating food resources (Jillson…...

    [...]

Journal ArticleDOI
TL;DR: Experimental dissection of empirical systems is providing important insights into the details of the drivers of demographic responses and therefore dynamics and should also stimulate theory that incorporates relevant biological mechanism.
Abstract: Population dynamics result from the interplay of density-independent and density-dependent processes. Understanding this interplay is important, especially for being able to predict near-term population trajectories for management. In recent years, the study of model systems—experimental, observational and theoretical—has shed considerable light on the way that the both density-dependent and -independent aspects of the environment affect population dynamics via impacting on the organism's life history and therefore demography. These model-based approaches suggest that (i) individuals in different states differ in their demographic performance, (ii) these differences generate structure that can fluctuate independently of current total population size and so can influence the dynamics in important ways, (iii) individuals are strongly affected by both current and past environments, even when the past environments may be in previous generations and (iv) dynamics are typically complex and transient due to environmental noise perturbing complex population structures. For understanding population dynamics of any given system, we suggest that ‘the devil is in the detail’. Experimental dissection of empirical systems is providing important insights into the details of the drivers of demographic responses and therefore dynamics and should also stimulate theory that incorporates relevant biological mechanism.

241 citations


Cites background from "Multiple attractors, saddles, and p..."

  • ...…system between different dynamical attractors; such stochastic switching between attractors can also be seen with less complex, periodic dynamics (Henson et al. 1999, 2002, 2003; Cushing et al. 2003; King et al. 2004; Costantino et al. 2005) as well as in disease dynamics (Keeling et al. 2001;…...

    [...]

Journal ArticleDOI
TL;DR: In this article, a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium.
Abstract: A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, non- linear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attri- buting the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic vari- ability and obtained conditional least-squares estimates for the model parameters. We gen- erated 2000 ''bootstrapped data sets'' with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distri- butions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, in- cluding treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.

192 citations


Cites background from "Multiple attractors, saddles, and p..."

  • ...…approaches (Costantino et al. 1995, 1997, 1998, Dennis et al. 1995, 1997, Ellner and Turchin 1995, Begon et al. 1996, Stenseth et al. 1996, Higgens et al. 1997, Leirs et al. 1997, Bjornstad et al. 1998, Cushing et al. 1998a, b, Finkenstadt et al. 1998, Dixon et al. 1999, Henson et al. 1999)....

    [...]

  • ...For example, noise may cause a cycle to shift phase (Henson et al. 1998) or, in a regime with multiple deterministic attractors, may cause an orbit to jump from one cyclic attractor to another (Henson et al. 1999)....

    [...]

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

References
More filters
Journal ArticleDOI
TL;DR: For example, it has been shown that the smaller the population, the more susceptible it is to extinction from various causes as mentioned in this paper, and therefore, conservation efforts have been and will continue to be at the single species level.
Abstract: Many species cannot survive in mandominated habitats. Reserves of essentially undisturbed habitat are necessary if such species are to survive in the wild. Aside from increased efforts to accelerate habitat acquisition for such species, the most pressing need facing conservationists is development of a predictive understanding of the relationship between a population's size and its chances of extinction. Biologists have long known that the smaller the population, the more susceptible it is to extinction from various causes. During the current era of heightened competition for use of the world's remaining wildlands, this qualitative understanding is of limited utility to conservation and natural resource planners. The old adage that "the bigger the reserve, the better" must be replaced with more precise prescriptions for how much land is enough to achieve conservation objectives. Efforts at making such determinations have been clouded by inconsistencies in the focus on the unit to be preserved (population, species, community, ecosystem) and lack of an explicit definition of what constitutes successful preservation (persistence for 10, 100, 1000 years, etc.). The intricate interdependencies of living things dictate that conservation efforts be focused on the community and ecosystem level. Unfortunately, the very magnitude of complexity of these systems makes such efforts difficult. Moreover, certain species are more sensitive than others to changing conditions and begin to decline prior to any noticeable degradation of the community to which they belong. Consequently, conservation efforts have been and, in many cases, will continue to be at the singlespecies level. Many species currently in jeopardy are large-bodied and/or specialized, two characteristics that usually

1,880 citations


"Multiple attractors, saddles, and p..." refers background in this paper

  • ...In particular, two broad classes of stochastic mechanisms important to populations have been widely discussed: environmental stochasticity and demographic stochasticity (May, 1974; Shaffer, 1981)....

    [...]

Journal ArticleDOI
01 Oct 1977-Nature
TL;DR: This review discusses how alternate stable states can arise in simple 1- and 2-species systems, and applies these ideas to grazing systems, to insect pests, and to some human host–parasite systems.
Abstract: Theory and observation indicate that natural multi-species assemblies of plants and animals are likely to possess several different equilibrium points. This review discusses how alternate stable states can arise in simple 1- and 2-species systems, and applies these ideas to grazing systems, to insect pests, and to some human host–parasite systems.

1,508 citations


"Multiple attractors, saddles, and p..." refers background in this paper

  • ...The prospect of multiple stable states has long been known to be a prediction of various nonlinear population models (May, 1977), but to date few convincing examples of multiple stable states have been documented (Petraitis and Latham, 1999)....

    [...]

Journal ArticleDOI
15 Nov 1974-Science
TL;DR: This paper presents a dynamical regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur.
Abstract: Some of the simplest nonlinear difference equations describing the growth of biological populations with nonoverlapping generations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between 2 population points, to stable cycles with 4, 8, 16, . . . points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized analyses; its existence in such fully deterministic nonlinear difference equations is a fact of considerable mathematical and ecological interest.

1,456 citations


"Multiple attractors, saddles, and p..." refers background in this paper

  • ...In particular, two broad classes of stochastic mechanisms important to populations have been widely discussed: environmental stochasticity and demographic stochasticity (May, 1974; Shaffer, 1981)....

    [...]

Book
01 Jan 1982
TL;DR: The classic textbook Modeling Fluctuating Populations, originally published in 1982, is a classic textbook because primarily it takes a distinctive approach to population dynamics, by emphasizing from the earliest chapters that all populations fluctuate continuously as discussed by the authors.
Abstract: Modelling Fluctuating Populations, originally published in 1982, is a classic textbook because primarily, this book takes a distinctive approach to population dynamics, by emphasizing from the earliest chapters that all populations fluctuate continuously. Traditional themes in theoretical ecology such as equilibrium and population stability are linked to analyses of the response of a population to environmental fluctuations and to extinction probabilities. Thus, the book's approach confronts head-on one common criticism of simple ecological models - the mismatch between the mathematical mechanisms studied and the questions of top ecological concern. Secondly, the book demonstrates the power of techniques based on linear mathematics.

831 citations

Book
01 Jan 1990
TL;DR: In this article, a list of symbols and notation for birth-death processes is presented. But this list is limited to simple birth death processes and does not cover the complexity of complex processes such as predator-prey systems.
Abstract: Preface A list of symbols and notation 1. Introductory remarks 2. Simple birth-death processes 3. General birth-death processes 4. Time-lag models of population growth 5. Competition processes 6. Predator-prey processes 7. Spatial predator-prey systems 8. Fluctuating environments 9. Spatial population dynamics 10. Epidemic processes 11. Linear and branching architectures References Author index Subject index.

725 citations