Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

TL;DR

This paper investigates how nonlocal interactions influence pattern formation and phase transitions in a single-species metapopulation model, revealing conditions for stable patterns and bifurcations as dispersal rates vary.

Contribution

It introduces a novel single-species metapopulation model with nonlocal interactions and analyzes pattern formation through stability and bifurcation analysis.

Findings

Patterns form when dispersal rate is low and interaction kernel meets certain conditions.

Traveling and stationary wave patterns emerge near critical dispersal rates.

Pattern amplitude increases as dispersal rate decreases, with thresholds for initial condition amplitude.

Abstract

The presence of one or more species at some spatial locations but not others is a central matter in ecology. This phenomenon is related to ecological pattern formation. Nonlocal interactions can be considered as one of the mechanisms causing such a phenomenon. We propose a single-species, continuous time metapopulation model taking nonlocal interactions into account. Discrete probability kernels are used to model these interactions in a patchy environment. A linear stability analysis of the model shows that solutions to this equation exhibit pattern formation if the dispersal rate of the species is sufficiently small and the discrete interaction kernel satisfies certain conditions. We numerically observe that traveling and stationary wave type patterns arise near critical dispersal rate. We use weakly nonlinear analysis to better understand the behavior of formed patterns. We show that observed patterns arise through both supercritical and subcritical bifurcations from spatially homogeneous steady state. Moreover, we observe that as the dispersal rate decreases, amplitude of the patterns increases. For discontinuous transitions to instability, we also show that there exists a threshold for the amplitude of the initial condition, above which pattern formation is observed.