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When can a population spreading across sink habitats persist ?

Michel Benaim, Claude Lobry, Tewfik Sari, Edouard Strickler

Abstract

We consider populations with time-varying growth rates living in sinks. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. We provide a mathematical analysis of this surprising phenomenon, in the context of a deterministic model with periodic variation of growth rates and non-symmetric migration which are assumed to be piecewise continuous. We also consider a stochastic model with random variation of growth rates and migration. This work extends existing results of the literature on the DIG effects obtained for periodic continuous growth rates and time independent symmetric migration.

When can a population spreading across sink habitats persist ?

Abstract

We consider populations with time-varying growth rates living in sinks. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. We provide a mathematical analysis of this surprising phenomenon, in the context of a deterministic model with periodic variation of growth rates and non-symmetric migration which are assumed to be piecewise continuous. We also consider a stochastic model with random variation of growth rates and migration. This work extends existing results of the literature on the DIG effects obtained for periodic continuous growth rates and time independent symmetric migration.
Paper Structure (47 sections, 27 theorems, 182 equations, 15 figures, 4 tables)

This paper contains 47 sections, 27 theorems, 182 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Periodic environment
  3. The model
  4. The growth rate
  5. The DIG threshold
  6. Results
  7. Definitions and notations
  8. Asymptotics of $\Lambda(m,T)$ for large or small $m$ and $T$
  9. The DIG phenomenon
  10. Cooperative linear $T$-periodic systems
  11. An integral formula for the growth rate
  12. Time independent migration
  13. The monotonicity of $\Lambda(m,T)$ with respect to $T$
  14. Stochastic environment
  15. Examples and numerical illustrations
  16. ...and 32 more sections

Key Result

Proposition 1

Assume that Hypotheses H1 and H2 are satisfied. Suppose $m>0$ and $T>0$. Let $\mu(m,T)$ be the Perron root of the monodromy matrix $\Phi(T)$ of eq3. If $x(t)$ is a solution of eq3 such that $x(0)>0$, then for all $i$ The function $\Lambda$ is analytic in $m$ and $T$.

Figures (15)

  • Figure 1: The definition of $\Lambda(m,T)$ and its limits when $T$ tends to 0 or $\infty$ and/or $m$ tends to 0 or $\infty$.
  • Figure 2: (a) The graph of $(m,T)\mapsto \Lambda(m,T)$. (b) The set $\Lambda(m,T)=0$. (c) Graphs of $m\mapsto \Lambda(m,T)$ with the indicated values of $T$. (d) Graphs of $T\mapsto \Lambda(m,T)$ with the indicated values of $m$. Here we used the two patch model corresponding to the matrices \ref{['AB1']} and $\alpha=0.5$.
  • Figure 3: The set where $\Lambda(m,1/\nu)>0$ in the $(m,\nu)$ parameter-plane. (a) Parameters values of \ref{['AB1']}. (b) Parameters values of \ref{['AB2S']}. (c) Parameters values of \ref{['ABm_star_infini']}
  • Figure 4: The graph of $(m,T)\mapsto \Lambda(m,T)$ corresponding to the matrices \ref{['ABCmatrix']}, seen from left (a) and right (b), showing the non monotonicity of $T\mapsto \Lambda(m,T)$.
  • Figure 5: (a) The set $\Lambda(m,T)=0$. (b) The set $\Lambda(m,\nu)=0$. Here we use the parameter values of \ref{['ABCmatrix']} ($m^*=1.764$)
  • ...and 10 more figures

Theorems & Definitions (67)

  • Proposition 1
  • proof
  • Remark 1
  • Definition 1
  • Definition 2
  • Theorem 2
  • proof
  • Remark 2
  • Lemma 3
  • Remark 3
  • ...and 57 more