Asymptotic regimes of an integro-difference equation with discontinuous kernel

TL;DR

This paper studies an integral equation modeling population dynamics in patchy landscapes with discontinuous kernels, establishing conditions for existence, uniqueness, and extinction of stationary states.

Contribution

It provides new theoretical results on the existence, uniqueness, and extinction criteria for solutions of integro-difference equations with discontinuous kernels.

Findings

Existence and uniqueness of stationary states under certain conditions.

Criteria for population extinction based on eigenvalue analysis.

Analysis of asymptotic regimes in patchy landscapes.

Abstract

This paper is concerned with an integral equation that models discrete time dynamics of a population in a patchy landscape. The patches in the domain are reflected through the discontinuity of the kernel of the integral operator at a finite number of points in the whole domain. We prove the existence and uniqueness of a stationary state under certain assumptions on the principal eigenvalue of the linearized integral operator and the growth term as well. We also derive criteria under which the population undergoes extinction (in which case the stationary solution is 0 everywhere).