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

Hening, Alexandru; Nguyen, Dang H.; Yin, George

doi:10.1007/s00285-017-1153-2

Cited by 46 publications

(36 citation statements)

References 60 publications

(137 reference statements)

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“…These frequency changes within the population may alter the rate at which the population grows creating a feedback between the population density and the trait distribution (the internal variable) [Vincent and Brown, 2005, Schoener, 2011. Similarly, for populations structured by age or space, the distribution of ages or spatial locations are internal variables that often generate feedbacks with the total population density [Chesson, 2000, Evans et al, 2013, Hening et al, 2018.…”

Section: Introductionmentioning

confidence: 99%

Persistence and extinction for stochastic ecological models with internal and external variables

2019

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The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. To understand this interplay, we prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities i.e. stochastic persistence, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, in the case of persistence, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight several challenges for future research. 0 * Corresponding author: sschreiber@ucdavis.edu 1 arXiv:1808.07888v3 [q-bio.PE]

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Section: Introductionmentioning

confidence: 99%

Persistence and extinction for stochastic ecological models with internal and external variables

2019

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“…where the last inequality comes from (31). Thus, choosing C = K 1−2e −ρt * , one has W d (δ z P t , δzP t ) ≤ 1 + e −ρt − 2e −ρt * ≤ 1 − e −ρt * , proving A2 with ε = e −ρt * .…”

Section: Now By Feller Continuity Of P T and Compactness Ofmentioning

confidence: 66%

Random switching between vector fields having a common zero

Benaı̈m¹,

Strickler²

2019

Ann. Appl. Probab.

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Let E be a finite set, {F i } i∈E a family of vector fields on R d leaving positively invariant a compact set M and having a common zero p ∈ M. We consider a piecewise deterministic Markov process (X, I) on M × E defined byẊ t = F It (X t ) where I is a jump process controlled by X :We show that the behavior of (X, I) is mainly determined by the behavior of the linearized process (Y, J) whereẎ t = A Jt Y t , A i is the Jacobian matrix of F i at p and J is the jump process with rates (a ij (p)). We introduce two quantities Λ − and Λ + respectively defined as the minimal (respectively maximal) growth rate of Y t , where the minimum (respectively maximum) is taken over all the ergodic measures of the angular process (Θ, J) with Θ t = Yt Yt . It is shown that Λ + coincides with the top Lyapunov exponent (in the sense of ergodic theory) of (Y, J) and that under general assumptions Λ − = Λ + . We then prove that, under certain irreducibility conditions, X t → p exponentially fast when Λ + < 0 and (X, I) converges in distribution at an exponential rate toward a (unique) invariant measure supported by M \ {p} × E when Λ − > 0. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.

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“…In order to allow for environmental fluctuations and their effect on the persistence or extinction of species one approach is to study stochastic differential equations ( [18,36,25,22,24,23]). The other possible approach is to look at stochastic equations driven by a Markov chain.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On a Predator-Prey System with Random Switching that Never Converges to its Equilibrium

Hening¹,

Strickler²

2019

SIAM J. Math. Anal.

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We study the dynamics of a predator-prey system in a random environment. The dynamics evolves according to a deterministic Lotka-Volterra system for an exponential random time after which it switches to a different deterministic Lotka-Volterra system. This switching procedure is then repeated. The resulting process is a Piecewise Deterministic Markov Process (PDMP). In the case when the equilibrium points of the two deterministic Lotka-Volterra systems coincide we show that almost surely the trajectory does not converge to the common deterministic equilibrium. Instead, with probability one, the densities of the prey and the predator oscillate between 0 and ∞. This proves a conjecture of Takeuchi et al (J. Math. Anal. Appl 2006).The proof of the conjecture is a corollary of a result we prove about linear switched systems. Assume (Y t , I t ) is a PDMP that evolves according to dYt dt = A It Y t where A 0 , A 1 are 2 × 2 matrices and I t is a Markov chain on {0, 1} with transition rates k 0 , k 1 > 0. If the matrices A 0 and A 1 are not proportional and are of the formwith α 2 i + β i γ i < 0, then there exists λ > 0 such that lim t→∞ log Yt t = λ.

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

Cited by 46 publications

References 60 publications

Persistence and extinction for stochastic ecological models with internal and external variables

Persistence and extinction for stochastic ecological models with internal and external variables

Random switching between vector fields having a common zero

On a Predator-Prey System with Random Switching that Never Converges to its Equilibrium

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