Boundary perturbations and steady states of structured populations

Abstract

In this work we establish conditions which guarantee the existence of (strictly) positive steady states of a nonlinear structured population model. In our framework, the steady state formulation amounts to recasting the nonlinear problem as a family of eigenvalue problems, combined with a fixed point problem. Amongst other things, our formulation requires us to control the growth behaviour of the spectral bound of a family of linear operators along positive rays. For the specific class of model we consider here this presents a considerable challenge. We are going to show that the spectral bound of the family of operators, arising from the steady state formulation, can be controlled by perturbations in the domain of the generators (only). These new boundary perturbation results are particularly important for models exhibiting fertility controlled dynamics. As an important by-product of the application of the boundary perturbation results we employ here, we recover (using a recent theorem by H. R. Thieme) the familiar net reproduction number (or function) for models with single state at birth, which include for example the classic McKendrick (linear) and Gurtin-McCamy (non-linear) age-structured models.