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The goal of this note is to present a general approach to define the netreproduction function for a large class of nonlinear physiologically structuredpopulation models. In particular, we are going to show that this can beachieved in a natural way by reformulating a nonlinear problem as a family oflinear ones; each of the linear problems describing the evolution of thepopulation in a different, but constant environment. The reformulation of anonlinear population model as a family of linear ones is a new approach, andprovides an elegant way to study qualitative questions, for example theexistence of positive steady states. To define the net reproduction number forany fixed (constant) environment, i.e. for the linear models, we use a fairlyrecent spectral theoretic result, which characterizes the connection betweenthe spectral bound of an unbounded operator and the spectral radius of acorresponding bounded operator. For nonlinear models, varying the environmentnaturally leads to a net reproduction function.

Net reproduction functions for nonlinear structured population models