扰动拟正矩阵的谱单调性及其在种群动力学中的应用
Spectral Monotonicity of Perturbed Quasi-positive Matrices with Applications in Population Dynamics
摘要：
种群动力学中的门限值可以用矩阵的谱界表示，它决定了种群的持久性和灭绝性。对于方阵$μA + Q$，其中$A$是描述异质环境中斑块间种群扩散的拟正矩阵，$Q$是描述斑块内种群动态的对角矩阵，通过两种方法建立了其谱界关于扩散速度/耦合强度/行进频率$μ$的单调性.第一种方法是利用基于基尔霍夫矩阵树定理的图论方法的解析推导;第二种方法采用矩阵理论和复分析参数的Collatz Wielandt公式。结果表明，我们建立的结果是Karlin-Altenberg定理的一个稍微加强的版本，该定理是在研究进化生物学中的还原原理和斑块景观中的进化扩散时独立发现的。尽管如此，我们的研究结果提供了一个新的和有效的方法在复杂的生物系统的稳定性分析在异质环境。我们说明了这一点，我们的结果应用到著名的生态模型的单物种，捕食者-猎物和竞争，和一个流行病模型的易感-感染-易感（SIS）型。我们成功地解决了人口动力学文献中的一些开放问题。
英文摘要：
Threshold values in population dynamics can be formulated as spectral bounds of matrices, determining the dichotomy of population persistence and extinction. For a square matrix $μA + Q$, where $A$ is a quasi-positive matrix describing population dispersal among patches in a heterogeneous environment and $Q$ is a diagonal matrix encoding within-patch population dynamics, the monotonicy of its spectral bound with respect to dispersal speed/coupling strength/travel frequency $μ$ is established via two methods. The first method is an analytic derivation utilizing a graph-theoretic approach based on Kirchhoff's Matrix-Tree Theorem; the second method employs Collatz-Wielandt formula from matrix theory and complex analysis arguments. It turns out that our established result is a slightly strengthen version of Karlin-Altenberg's Theorem, which has previously been discovered independently while investigating reduction principle in evolution biology and evolution dispersal in patchy landscapes. Nevertheless, our result provides a new and effective approach in stability analysis of complex biological systems in a heterogeneous environment. We illustrate this by applying our result to well-known ecological models of single species, predator-prey and competition, and an epidemiological model of susceptible-infected-susceptible (SIS) type. We successfully solve some open problems in the literature of population dynamics.
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