Numerical dynamics of integrodifference equations: Periodic solutions and invariant manifolds in $C^\alpha(\Omega)$

TL;DR

This paper investigates the numerical dynamics of integrodifference equations in ecological models, focusing on how Nyström discretization affects periodic solutions and invariant manifolds, with proven convergence and persistence results.

Contribution

It provides a rigorous analysis of the effects of Nyström discretization on the local dynamics of periodic integrodifference equations with H"older continuous functions.

Findings

Proves persistence of hyperbolic periodic solutions.

Establishes convergence of stable and unstable manifolds.

Demonstrates the convergence order aligns with quadrature method.

Abstract

Integrodifference equations are versatile models in theoretical ecology for the spatial dispersal of species evolving in non-overlapping generations. The dynamics of these infinite-dimensional discrete dynamical systems is often illustrated using computational simulations. This paper studies the effect of Nystr\"om discretization to the local dynamics of periodic integrodifference equations with H\"older continuous functions over a compact domain as state space. We prove persistence and convergence for hyperbolic periodic solutions and their associated stable and unstable manifolds respecting the convergence order of the quadrature/cubature method.