空间异质环境中的随机种群增长
Stochastic population growth in spatially heterogeneous environments
摘要：
经典生态学理论预测，环境随机性通过降低人口的人均增长率来增加灭绝风险。为了理解环境随机性、空间异质性和扩散对种群增长的相互作用，我们研究了$n$斑块中种群丰度的以下模型：给定$X_t=x$的$X_{t+dt}$的条件定律是这样的：当$dt$很小时，$X_{t+dt}^i-X_t^i$的条件平均值近似为$[x^iμ_i+\sum_j(x^j D_{ji}-x^i D_{ij})]dt$，其中，$X_t^i$和$μ_i$分别是第$i$个斑块的丰度和人均增长率，$D_{ij}$是从第$i$个斑块到第$j$个斑块的扩散率，并且$X_{t+dt}^i-X_t^i$和$X_{t+dt}^j-X_t^j$的条件协方差近似为$x^i x^j σ_{ij}dt$。我们表明，这样一个空间扩展的人口，如果$S_t=(X_t^1+...+X_t^n)$是总人口丰度，然后$Y_t=X_t/S_t$，斑块比例的矢量，收敛于法律的随机矢量$Y_\infty$作为$t\to\infty$，和随机增长率$\lim_{t\to\infty}t^{-1}\log S_t$等于时空平均人均增长率$\sum_iμ_i\E[Y_\infty^i]$经历的人口减去一半的时空平均时间变化$\E[\sum_{i,j}σ_{ij}Y_\infty^i Y_\infty^j]$经历的人口。我们推导出分析结果的规律$Y_\infty$，找到选择的扩散机制$D$产生一个最佳的随机增长率为自由分散的人口，并调查的随机增长率的影响扩散率的约束。我们的研究结果提供了基本的见解“理想的自由”运动的不确定性，耦合汇人口的持久性，扩散率的演变，以及在保护生物学的单个大或几个小（SLOSS）的辩论。
英文摘要：
Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study the following model for population abundances in $n$ patches: the conditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small the conditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^iμ_i+\sum_j(x^j D_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $μ_i$ are the abundance and per capita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is the dispersal rate from the $i$-th to the $j$-th patch, and the conditional covariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^i x^j σ_{ij}dt$. We show for such a spatially extended population that if $S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$, the vector of patch proportions, converges in law to a random vector $Y_\infty$ as $t\to\infty$, and the stochastic growth rate $\lim_{t\to\infty}t^{-1}\log S_t$ equals the space-time average per-capita growth rate $\sum_iμ_i\E[Y_\infty^i]$ experienced by the population minus half of the space-time average temporal variation $\E[\sum_{i,j}σ_{ij}Y_\infty^i Y_\infty^j]$ experienced by the population. We derive analytic results for the law of $Y_\infty$, find which choice of the dispersal mechanism $D$ produces an optimal stochastic growth rate for a freely dispersing population, and investigate the effect on the stochastic growth rate of constraints on dispersal rates. Our results provide fundamental insights into "ideal free" movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology.
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