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.

摘要

我们考虑空间随机逻辑模型中的灭绝机制${\mathbb{R}}^d$（又名 Bolker-Pacala-Dieckmann-Law 空间种群模型）使用超出平均场方程的一阶扰动。在空间均匀的情况下（即当密度是非空间的并且协方差是平移不变的），我们表明随着时间趋于无穷大，扰动会收敛。这产生了稳态密度的一阶近似值。接下来，我们研究临界死亡率——确保种群灭绝的最小恒定死亡率——作为平均场缩放参数的函数$\epsilon >0$. 我们找到渐近展开的首项（如$\epsilon \to 0$) 的临界死亡率，这对于病例来说显然是不同的$d\ge 3$，d  = 2，并且d  = 1。