Random evolution in population dynamics

Abstract

We present a perturbative formalism to deal with linear random positive maps. We generalize the biological concept of the population growth rate when a Leslie matrix has random elements (i.e. characterizing the macroscopic disorder in the vital parameters). The dominant eigenvalue of which defines the asymptotic dynamics of the mean-value population vector state, is presented as the effective growth rate of a random Leslie model. The problem was reduced to the calculation of the smallest positive root z̃1 of the secular polynomial appearing in the general expression for the mean-value Green function 〈G(z)〉. This nontrivial polynomial can be obtained order by order in terms of a diagrammatic technique built with Terwiel's cumulants, which have carefully been identified in the present work. By understanding how this smallest positive root z̃1 = 1/λ̃1 depends on the model of disorder, one can link the asymptotic population dynamics with the statistical properties of the errors (mutations) in the vital parameters. This eigenvalue has the meaning of an effective PerronFrobenious eigenvalue for a random positive matrix. Analytical (exact and perturbative calculations) results are presented for several models of disorder.