Size-structured populations: immigration, (bi)stability and the net growth rate

TL;DR

This paper analyzes size-structured population models with immigration, demonstrating how external inflow influences stability, bifurcations, and potential bistability, with implications for population management.

Contribution

It introduces a framework linking linear stability to a generalized net reproduction function and explores nonlinear dynamics under immigration effects.

Findings

Linearised system governed by a quasicontraction semigroup

Stability determined by a generalized net reproduction function

Immigration can induce bistability and bifurcations in population equilibria

Abstract

We consider a class of physiologically structured population models, a first order nonlinear partial differential equation equipped with a nonlocal boundary condition, with a constant external inflow of individuals. We prove that the linearised system is governed by a quasicontraction semigroup. We also establish that linear stability of equilibrium solutions is governed by a generalized net reproduction function. In a special case of the model ingredients we discuss the nonlinear dynamics of the system when the spectral bound of the linearised operator equals zero, i.e. when linearisation does not decide stability. This allows us to demonstrate, through a concrete example, how immigration might be beneficial to the population. In particular, we show that from a nonlinearly unstable positive equilibrium a linearly stable and unstable pair of equilibria bifurcates. In fact, the linearised system exhibits bistability, for a certain range of values of the external inflow, induced potentially by All\'{e}e-effect.