Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback

TL;DR

This paper analyzes the long-term behavior of a size-structured cannibalism model with environmental feedback, providing conditions for stability, instability, and growth using spectral analysis of the linearized system.

Contribution

It introduces a novel asymptotic analysis framework for a size-structured cannibalism model with infinite-dimensional environmental feedback, including spectral characterization and stability conditions.

Findings

Conditions for exponential stability of the linearized system.

Characterization of the point spectrum in special cases.

Criteria for asynchronous exponential growth.

Abstract

In this work we consider a size-structured cannibalism model with the model ingredients (fertility, growth, and mortality rate) depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system, in particular on the effect of cannibalism on the long-term dynamics. To this end, we formally linearize the system about steady state and establish conditions in terms of the model ingredients which yield uniform exponential stability of the governing linear semigroup. We also show how the point spectrum of the linearized semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup.