Random Leslie matrices in population dynamics.

We generalize the concept of the population growth rate when a Leslie matrix has random elements (correlated or not), i.e., characterizing the disorder in the vital parameters. In general, we present a perturbative formalism to deal with linear non-negative random matrix difference equations, then the non-trivial effective eigenvalue of which defines the long-time asymptotic dynamics of the mean-value population vector state is presented as the effective growth rate. This effective eigenvalue is calculated from the smallest positive root of a secular polynomial. Analytical (exact and perturbative calculations) results are presented for several models of disorder. In particular, a 3 × 3 numerical example is applied to study the effective growth rate characterizing the long-time dynamics of a biological population model. The present analysis is a perturbative method for finding the effective growth rate in cases when the vital parameters may have negative covariances across populations.