Spatially inhomogeneous two-cycles in an integrodifference equation

TL;DR

The work develops a rigorous, computer-assisted framework to prove the existence of spatially inhomogeneous 2-cycles in an integro-difference equation with a Laplace kernel and logistic growth, by recasting the problem as a symmetry-reduced connecting orbit in a 4D ODE. It combines Banach-algebra and series methods with the parameterization method to construct a local stable manifold and then solve a projected boundary-value problem via Chebyshev series, all certified through Newton-Kantorovich-type bounds. The authors prove a logistic-growth 2-cycle and provide compelling numerical evidence and a roadmap to extend the proof to the Ricker case, along with spectral-stability indicators via an Evans-function framework. Together, these results advance a rigorous understanding of nontrivial spatial patterns in IDEs and showcase a constructive approach that integrates dynamical systems, functional analysis, and rigorous numerics with potential applicability to travelling 2-cycles and broader dispersal kernels.

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.

Figures (8)

• Figure 1: Two-cycle of \ref{['IDE1']}. In red, we have $\bar{N}(t)=N_0(t)$ and in blue $\bar{M}(t)=N_1(t)$. In black, horizontal lines corresponding to $n_\pm$.
• Figure 2: Plot of the three-dimensional projection of the approximate local stable manifold at $\tilde{x}^{(+)}$ and local unstable manifold at $\tilde{x}^{(-)}$. The fourth dimension is expressed in the color of the plot, specified by the color bar. These are explicitly given by ${P}([-1,1]^2)$ and $R{P}([-1,1]^2)$, respectively. This figure, and all other figures present in this paper, were created using the GLMakie.jl package Danisch2021.
• Figure 3: Solution to the projected BVP when $\sigma=10$ and $\rho=2.2$. Plot of the four coordinates of $\bar{\Gamma}(t)$ for $t \in [0, 2\bar{L}]$. One notices that $\bar{\Gamma}(0) \approx R\bar{\Gamma}(0)$ as expected.
• Figure 4: Plot of $\bar{N}(t), \bar{M}(t)$ for $t \ge 0$. $\bar{N}(t)$ appears as a solid line while $\bar{M}(t)$ appears as a dotted line. The color and legend is to precisely show how these functions are constructed from their pieces. For negative values of $t$, recall the symmetry presented above.
• Figure 5: Left: Plot of approximate stable and unstable manifolds along with the connection between the two. In red, we have the two fixed points of interest. The colored sheets are the manifolds and the colored line is the connection between them. The four-dimensional objects are projected to $\mathbb{R}^3$ and the fourth dimension is represented by color. Right: The corresponding spatially inhomogeneous two-cycle of \ref{['IDE1']}. In red, we have $\bar{N}(t)=N_0(t)$ and in blue $\bar{M}(t)=N_1(t)$. In black, horizontal lines corresponding to $n_\pm$.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Remark 2.6