相互作用结构种群的鲁棒持久性
Robust permanence for interacting structured populations
摘要：
相互作用的结构化群体的动力学可以由$\frac{dx_i}{dt}= A_i (x)x_i$建模，其中$x_i\in \R^{n_i}$、$x=(x_1,\dots,x_k)$和$A_i(x)$是具有非负非对角项的矩阵。这些模型是永久的，如果存在一个积极的全球吸引子和鲁棒永久的，如果他们保持永久的扰动$A_i(x)$。利用$A_i(x)$关于不变测度$μ$的主导李雅普诺夫指数$λ_i(μ)$，导出了鲁棒持久的充分必要条件.必要条件要求对所有在非负锥边界上有支撑的遍历测度都有$\max_i λ_i(μ)>0$。充分条件要求边界允许Morse分解，使得对于所有不变测度$μ$，$\max_i λ_i(μ)>0$由Morse分解的分量支持。当Morse分量是公理A，唯一遍历，或支持所有，但一个人口，必要和充分条件是等价的。空间生态学，流行病学和基因网络的应用。
英文摘要：
The dynamics of interacting structured populations can be modeled by $\frac{dx_i}{dt}= A_i (x)x_i$ where $x_i\in \R^{n_i}$, $x=(x_1,\dots,x_k)$, and $A_i(x)$ are matrices with non-negative off-diagonal entries. These models are permanent if there exists a positive global attractor and are robustly permanent if they remain permanent following perturbations of $A_i(x)$. Necessary and sufficient conditions for robust permanence are derived using dominant Lyapunov exponents $λ_i(μ)$ of the $A_i(x)$ with respect to invariant measures $μ$. The necessary condition requires $\max_i λ_i(μ)>0$ for all ergodic measures with support in the boundary of the non-negative cone. The sufficient condition requires that the boundary admits a Morse decomposition such that $\max_i λ_i(μ)>0$ for all invariant measures $μ$ supported by a component of the Morse decomposition. When the Morse components are Axiom A, uniquely ergodic, or support all but one population, the necessary and sufficient conditions are equivalent. Applications to spatial ecology, epidemiology, and gene networks are given.
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