4,150 research outputs found

    Persistence in fluctuating environments

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    Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations' invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka-Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.Comment: 25 page

    A metapopulation model with Markovian landscape dynamics

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    We study a variant of Hanski's incidence function model that allows habitat patch characteristics to vary over time following a Markov process. The widely studied case where patches are classified as either suitable or unsuitable is included as a special case. For large metapopulations, we determine a recursion for the probability that a given habitat patch is occupied. This recursion enables us to clarify the role of landscape dynamics in the survival of a metapopulation. In particular, we show that landscape dynamics affects the persistence and equilibrium level of the metapopulation primarily through its effect on the distribution of a local population's life span.Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

    Persistence for stochastic difference equations: A mini-review

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    Understanding under what conditions populations, whether they be plants, animals, or viral particles, persist is an issue of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic forces is the construction and analysis of stochastic difference equations Xt+1=F(Xt,ξt+1)X_{t+1}=F(X_t,\xi_{t+1}) where Xt∈RkX_t \in \R^k represents the state of the populations and ξ1,ξ2,...\xi_1,\xi_2,... is a sequence of random variables representing environmental stochasticity. In the analysis of these stochastic models, many theoretical population biologists are interested in whether the models are bounded and persistent. Here, boundedness asserts that asymptotically XtX_t tends to remain in compact sets. In contrast, persistence requires that XtX_t tends to be "repelled" by some "extinction set" S0⊂RkS_0\subset \R^k. Here, results on both of these proprieties are reviewed for single species, multiple species, and structured population models. The results are illustrated with applications to stochastic versions of the Hassell and Ricker single species models, Ricker, Beverton-Holt, lottery models of competition, and lottery models of rock-paper-scissor games. A variety of conjectures and suggestions for future research are presented.Comment: Accepted for publication in the Journal of Difference Equations and Application

    Stochastic population growth in spatially heterogeneous environments: The density-dependent case

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    This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout nn patches whose population abundances are modelled as the solutions of a system of nonlinear stochastic differential equations living on [0,∞)n[0,\infty)^n. We prove that rr, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter rr can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if r>0r>0, the population abundances converge polynomially fast to a unique invariant probability measure on (0,∞)n(0,\infty)^n, while when r<0r<0, the population abundances of the patches converge almost surely to 00 exponentially fast. This generalizes and extends the results of Evans et al (2014 J. Math. Biol.) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. As an example we fully analyze the two-patch case, n=2n=2, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al.Comment: 43 pages, 1 figure, edited according to the suggestion of the referees, to appear in Journal of Mathematical Biolog

    Unrelated toxin-antitoxin systems cooperate to induce persistence.

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    Persisters are drug-tolerant bacteria that account for the majority of bacterial infections. They are not mutants, rather, they are slow-growing cells in an otherwise normally growing population. It is known that the frequency of persisters in a population is correlated with the number of toxin-antitoxin systems in the organism. Our previous work provided a mechanistic link between the two by showing how multiple toxin-antitoxin systems, which are present in nearly all bacteria, can cooperate to induce bistable toxin concentrations that result in a heterogeneous population of slow- and fast-growing cells. As such, the slow-growing persisters are a bet-hedging subpopulation maintained under normal conditions. For technical reasons, the model assumed that the kinetic parameters of the various toxin-antitoxin systems in the cell are identical, but experimental data indicate that they differ, sometimes dramatically. Thus, a critical question remains: whether toxin-antitoxin systems from the diverse families, often found together in a cell, with significantly different kinetics, can cooperate in a similar manner. Here, we characterize the interaction of toxin-antitoxin systems from many families that are unrelated and kinetically diverse, and identify the essential determinant for their cooperation. The generic architecture of toxin-antitoxin systems provides the potential for bistability, and our results show that even when they do not exhibit bistability alone, unrelated systems can be coupled by the growth rate to create a strongly bistable, hysteretic switch between normal (fast-growing) and persistent (slow-growing) states. Different combinations of kinetic parameters can produce similar toxic switching thresholds, and the proximity of the thresholds is the primary determinant of bistability. Stochastic fluctuations can spontaneously switch all of the toxin-antitoxin systems in a cell at once. The spontaneous switch creates a heterogeneous population of growing and non-growing cells, typical of persisters, that exist under normal conditions, rather than only as an induced response. The frequency of persisters in the population can be tuned for a particular environmental niche by mixing and matching unrelated systems via mutation, horizontal gene transfer and selection

    Spatial models of metapopulations and benthic communities in patchy environments

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    Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September 2000The distribution of organisms in space has important consequences for the function and structure of ecological systems. Such distributions are often referred to as patchy, and a patch-based approach to modeling ecosystem dynamics has become a major research focus. These models have been used to explore a wide range of questions concerning population, metapopulation, community, and landscape ecology, in both terrestrial and aquatic systems. In this dissertation I develop and analyze a series of spatial models to study the dynamics of metapopulations and marine benthic communities in patchy environments. All the models have the form of a discrete-time Markov chain, and assume that the landscape is composed of discrete patches, each of which is in one of a number of possible states. The state of a patch is determined by the presence of an individual of a given species, a local population, or a group of species, depending on the spatial scale of the model. The research is organized into two main parts as follows. In the first part, I present an analysis of the effects of habitat destruction on metapopulation persistence. Theoretical studies have already shown that a metapopulation goes extinct when the fraction of suitable patches in the landscape falls below a critical threshold (the so called extinction threshold). This result has become a paradigm in conservation biology and several models have been developed to calculate extinction thresholds for endangered species. These models, however, generally do not take into account the spatial arrangement of habitat destruction, or the actual size of the landscape. To investigate how the spatial structure of habitat destruction affects persistence, I compare the behavior of two models: a spatially implicit patch-occupancy model (which recreates the extinction patterns found in other models) and a spatially explicit cellular automaton (CA) model. In the CA, I use fractal arrangements of suitable and unsuitable patches to simulate habitat destruction and show that the extinction threshold depends on the fractal dimension of the landscape. To investigate how habitat destruction affects persistence in finite landscapes , I develop and analyze a chain-binomial metapopulation (CBM) model. This model predicts the expected extinction time of a metapopulation as a function of the number of patches in the landscape and the number of those patches that are suitable for the population. The CBM model shows that the expected time to extinction decreases greater than exponentially as suitable patches are destroyed. I also describe a statistical method for estimating parameters for the CBM model in order to evaluate metapopulation viability in real landscapes. In the second part, I develop and analyze a series of Markov chain models for a rocky subtidal community in the Gulf of Maine. Data for the model comes from ten permanent quadrats (located on Ammen Rock Pinnacle at 30 meters depth) monitored over an 8-year period (1986-1994). I first parameterize a linear (homogenous) Markov chain model from the data set and analyze it using an array of novel techniques, including a compression algorithm to classify species into functional groups, a set of measures from stochastic process theory to characterize successional patterns, sensitivity analyses to predict how changes in various ecological processes effect community composition, and a method for simulating species removal to identify keystone species. I then explore the effects of time and space on successional patterns using log-linear analysis, and show that transition probabilities vary significantly across small spatial scales and over yearly time intervals. I examine the implications of these findings for predicting equilibrium species abundances and for characterizing the transient dynamics of the community. Finally, I develop a nonlinear Markov chain for the rocky subtidal community. The model is parameterized using maximum likelihood methods to estimate density-dependent transition probabilities. I analyze the best fitting models to study the effects of nonlinear species interactions on community dynamics, and to identify multiple stable states in the subtidal system.This work was supported by the Office of Naval Research and the National Science Foundation through the following grants to Hal Caswell: ONR-URIP Grant NOOOl492- J-1527, NSF Grants DEB-9119420, DEB-95-27400, OCE-981267 and OCE-9302238
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