Extinction threshold in the spatial stochastic logistic model: space homogeneous case

ABSTRACT

We consider the extinction regime in the spatial stochastic logistic model in ${\mathbb{R}}^d$ (a.k.a. Bolker–Pacala–Dieckmann–Law model of spatial populations) using the first-order perturbation beyond the mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields the first-order approximation for the stationary density. Next, we study the critical mortality – the smallest constant death rate which ensures the extinction of the population – as a function of the mean-field scaling parameter $\epsilon >0$. We find the leading term of the asymptotic expansion (as $\epsilon \to 0$) of the critical mortality which is apparently different for the cases $d\ge 3$, d = 2, and d = 1.

Keywords:

• Extinction threshold
• spatial logistic model
• mean-field equation
• population density
• perturbation
• correlation function
• asymptotic behaviour

2010 Mathematics Subject Classifications:

• 34E10
• 92D25
• 34D10
• 58C15

Acknowledgments

The authors thank Otso Ovaskainen and Panu Somervuo for discussions that motivated the analyses presented in this paper.

Disclosure statement

No potential conflict of interest was reported by the author.