# Coexistence and extinction for stochastic Kolmogorov systems

@article{Hening2018CoexistenceAE,
title={Coexistence and extinction for stochastic Kolmogorov systems},
author={Alexandru Hening and Dang Hai Nguyen},
journal={The Annals of Applied Probability},
year={2018}
}
• Published 23 April 2017
• Mathematics, Biology
• The Annals of Applied Probability
In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of $n$ populations that live in a stochastic environment and which can interact nonlinearly (through… Expand
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#### References

SHOWING 1-10 OF 54 REFERENCES
Persistence in fluctuating environments
• Biology, Medicine
• Journal of mathematical biology
• 2011
A mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations is developed and it is illustrated that environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates. Expand
Stochastic population growth in spatially heterogeneous environments: the density-dependent case
• Mathematics, Biology
• Journal of mathematical biology
• 2018
It is proved that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise and the stochastic growth rate depends continuously on the coefficients. Expand
Persistence of structured populations in random environments.
• Biology, Medicine
• Theoretical population biology
• 2009
This work provides a general theory for persistence for density-dependent matrix models in random environments and shows that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space. Expand
Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction
• Mathematics, Medicine
• Journal of mathematical biology
• 2010
This work studies the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction, and generalizes in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. Expand
Coexistence in locally regulated competing populations and survival of branching annihilating random walk
• Mathematics
• 2007
We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analyticExpand
Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments
• Biology
• 2014
It is shown that adding environmental stochasticity results in an ESS that, when compared to the ESS for the corresponding model without stochasticsity, spends less time in patches with larger carrying capacities and possibly makes use of sink patches, thereby practicing a spatial form of bet hedging. Expand
Quasi-stationary distributions and diffusion models in population dynamics
• Mathematics
• 2007
In this paper, we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to $- \infty$ at the origin, and the diffusion to have anExpand
Persistence in Stochastic Lotka–Volterra Food Chains with Intraspecific Competition
• Mathematics, Medicine
• Bulletin of mathematical biology
• 2018
It is shown that one can classify, based on the invasion rates of the predators, which species go extinct and which converge to their unique invariant probability measure, and provides persistence/extinction criteria for food chains of length 4. Expand
Variable effort harvesting models in random environments: generalization to density-dependent noise intensities.
• C. Braumann
• Mathematics, Medicine
• Mathematical biosciences
• 2002
This paper generalizes the previous results to density-dependent positive noise intensities of very general form so that they also become independent from the way environmental fluctuations affect population growth rates. Expand
Invasibility and stochastic boundedness in monotonic competition models
• Mathematics
• 1989
We give necessary and sufficient conditions for stochastically bounded coexistence in a class of models for two species competing in a randomly varying environment. Coexistence is implied by mutualExpand