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

TL;DR

This paper investigates the existence of steady states in nonlinear structured population models with distributed recruitment by reformulating the problem as an integral equation and introducing a density-dependent net reproduction rate.

Contribution

It introduces a novel integral equation approach and explores the relationship between net reproduction rate and steady states in structured populations.

Findings

Reformulation of the steady state problem as an integral equation.

Introduction of a density-dependent net reproduction rate.

Discussion of finite rank approximation of recruitment operator.

Abstract

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 parametrized family of unbounded operators. The alternative approach, on which we focus here, is based on the reformulation of the problem 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 briefly discuss a third approach, which is based on the finite rank approximation of the recruitment operator.