空间不均匀环境中的随机种群增长：密度依赖的情况
Stochastic population growth in spatially heterogeneous environments: The density-dependent case
摘要：
这项工作是专门研究的动态结构人口的环境随机性，资源竞争，时空异质性和扩散的综合影响。人口分布在整个$n$补丁，其人口丰度建模为生活在$[0,\infty)^n$上的非线性随机微分方程系统的解决方案。我们证明了$r$，在没有竞争的情况下，总人口的随机增长率，决定了人口的长期行为。参数$r$可以表示为随机微分方程的相关线性化系统的李雅普诺夫指数。详细的分析表明，如果$r>0$，人口丰度多项式快速收敛到$(0,\infty)^n$上的一个唯一不变的概率测度，而当$r<0$，人口丰度斑块几乎必然收敛到$0$指数快速。这概括并扩展了Evans等人（2014 J. Math.Biol.）并证明了他们的一个优点。与最近的发展相比，我们的模型采用了非常普遍的密度依赖的增长率和竞争条款。此外，我们证明了持久性是强大的小，可能依赖于密度，扰动的增长率，扩散矩阵和协方差矩阵的环境噪声。我们的工作允许环境噪声驱动我们的系统退化。从生物学的角度来看，这是相关的，因为例如，不同斑块的环境可以完美地相关。作为一个例子，我们充分分析了两个补丁的情况下，$n=2$，并表明，随机增长率是一个递减函数的色散率。特别是，耦合两个汇补丁永远不会产生持久性，相反，埃文斯等人处理的非退化设置的结果。
英文摘要：
This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout $n$ patches whose population abundances are modelled as the solutions of a system of nonlinear stochastic differential equations living on $[0,\infty)^n$. We prove that $r$, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter $r$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $r>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\infty)^n$, while when $r<0$, the population abundances of the patches converge almost surely to $0$ exponentially fast. This generalizes and extends the results of Evans et al (2014 J. Math. Biol.) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. As an example we fully analyze the two-patch case, $n=2$, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al.
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