Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Analytical results and applications

TL;DR

This paper analyzes local bifurcations of periodic solutions in integrodifference equations used in ecology, providing explicit criteria for various bifurcation types and combining analytical and numerical methods for classification.

Contribution

It introduces explicit criteria for identifying bifurcation types in infinite-dimensional integrodifference equations, including classical scenarios, and applies these to ecological models.

Findings

Criteria for fold- and crossing curve-type bifurcations

Classification of transcritical, pitchfork, and flip bifurcations

Application of analytical and numerical methods to ecological models

Abstract

We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional discrete dynamical systems arise in theoretical ecology as models to describe the spatial dispersal of species having nonoverlapping generations. Our explicit criteria allow us to identify branchings of fold- and crossing curve-type, which include the classical transcritical-, pitchfork- and flip-scenario as special cases. Indeed, not only tools to detect qualitative changes in models from e.g. spatial ecology and related simulations are provided, but these critical transitions are also classified. In addition, the bifurcation behavior of various time-periodic integrodifference equations is investigated and illustrated. This requires a combination of analytical methods and numerical tools based on Nystr\"om discretization of the integral operators involved.