An abstract framework for a class of nonlocal structured population models: existence, uniqueness and stability of steady states

TL;DR

This paper develops an abstract framework for nonlocal structured population models, establishing conditions for existence, uniqueness, and stability of steady states, and providing counterexamples to highlight the limits of these conditions.

Contribution

It introduces a general abstract setting that unifies various population models and analyzes their well-posedness and stability properties within this framework.

Findings

Established conditions for existence and stability of steady states.

Provided counterexamples illustrating limits of stability conditions.

Applied framework to mutation--selection models with symmetric mutation operators.

Abstract

This paper is concerned with the study of a class of nonlinear nonlocal functional evolution problems defined in an abstract Banach algebra. We introduce an abstract functional setting that encompasses a wide range of structured population models appearing in biomathematical literature. Within this framework, we analyze the well-posedness of the Cauchy problem and the existence of stationary solutions in the positive cone of the Banach algebra. By reviewing a large number of approaches, we also derive conditions for the local and global stability of these stationary solutions. Additionally, we explore the limits of these conditions by exhibiting explicit counterexamples. In particular, for mutation--selection models with symmetric mutation operators, we uncover both sufficient conditions for existence, uniqueness and stability, and counterexamples to existence or stability.