Unique steady states for population models in a heterogeneous environment

TL;DR

This paper establishes conditions for the uniqueness of positive steady states in a population diffusion model within a heterogeneous environment, extending previous work by applying generalized logistic growth rates and Hamiltonian system techniques.

Contribution

It provides sufficient conditions for the uniqueness of steady states in a population model, utilizing Chicone's monotonicity approach for the period map in Hamiltonian systems.

Findings

Unique steady states are guaranteed under generalized logistic growth rates.

Sufficient conditions for steady state uniqueness are identified.

The approach leverages Hamiltonian system properties and monotonicity of the period map.

Abstract

We revisit a model proposed by Freedman etal in \cite{freedman} which describes the dynamics of a population diffusing in a patchy environment. From their work it is known that positive steady states exist for this model, but not whether they are unique. Here, we provide sufficient conditions guaranteeing that steady states are unique. These conditions are satisfied when the reaction rates are generalized logistic growth rates. Our proofs critically exploit Chicone's ideas in \cite{chicone}, which were used to establish that the period map associated to a continuous family of periodic solutions of certain planar Hamiltonian systems is monotone.