Two Novel proofs of Spectral Monotonicity of Perturbed Essentially Nonnegative Matrices with Applications in Population Dynamics
Shanshan Chen, Junping Shi, Zhisheng Shuai, Yixiang Wu
Abstract
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 $\rho A + Q$, where $A$ is an essentially nonnegative matrix describing population dispersal among patches in a heterogeneous environment and $Q$ is a real diagonal matrix encoding within-patch population dynamics, the monotonicity of its spectral bound with respect to dispersal rate/coupling strength/travel frequency $\rho$ has been established by Karlin and generalized by Altenberg while investigating the reduction principle in evolution biology and evolution dispersal in patchy landscapes. In this paper, we provide two new proofs rooted in our investigation of persistence in spatial population dynamics. The first one is an analytic derivation utilizing a graph-theoretic approach based on Kirchhoff's matrix-tree theorem; the second one employs the Collatz--Wielandt formula from matrix theory and complex analysis arguments. This monotonicity result has numerous applications in persistence and stability analysis of complex biological systems in heterogeneous environments. We illustrate this by applying it to well-known ecological models of single species, predator-prey, and competition.