Net reproduction functions for nonlinear structured population models

Our goal in this note is to present a general approach to define the net reproduction function/functional for a large class of nonlinear physiologically structured population models. In particular, we are going to show that this can be achieved in a natural way by reformulating a general nonlinear problem as a family of linear problems; each of the linear problems describing the evolution of the population in a different, but constant environment. The reformulation of a nonlinear population model as a family of linear ones is a new approach and provides an elegant way to study qualitative questions, in general. To define the net reproduction number for any fixed (constant) environment we use a fairly recent spectral theoretic result . For nonlinear models, varying the environment naturally leads to a net reproduction function or functional, depending on the dimension of the nonlinearity.