Spatially inhomogeneous two-cycles in an integrodifference equation

TL;DR

This paper proves the existence of spatially inhomogeneous 2-cycles in an integrodifference equation with logistic growth, connecting theoretical proof with numerical evidence of stability, extending prior numerical observations.

Contribution

It provides a rigorous proof of 2-cycles in an integrodifference model with logistic growth, linking numerical observations to theoretical validation.

Findings

Existence of 2-cycles in the model is proven mathematically.

Numerical results suggest the 2-cycles are spectrally stable.

The approach can potentially be applied to Ricker growth functions.

Abstract

In this work, we prove the existence of a 2-cycle in an integrodifference equation with a Laplace kernel and logistic growth function, connecting two non-trivial fixed points of the second iterate of the logistic map in the non-chaotic regime. This model was first studied by Kot (1992), and the 2-cycle we establish corresponds to one numerically observed by Bourgeois, Leblanc, and Lutscher (2018) for the Ricker growth function. We provide strong evidence that the 2-cycle for the Ricker growth function can be rigorously proven using a similar approach. Finally, we present numerical results indicating that both 2-cycles exhibit spectral stability.