On the net reproduction rate of continuous structured populations with distributed states at birth

We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least) in three different ways. One approach is to study spectral properties of a parametrised family of unbounded operators. The alternative approach, which we develop here, is based on the reformulation of the partial differential equation as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we discuss a third approach, which is based on a finite rank approximation of the recruitment operator.