Journal

Journal of Nonlinear Science

Abstract

Nonlocal aggregation-diffusion models, when coupled with a spatial map, can capture cognitive and memory-based influences on animal movement and population-level patterns. In this work, we study a one-dimensional reaction-diffusion-aggregation system in which a population’s spatiotemporal dynamics are tightly linked to a separate, dynamically updating map. Depending on the local population density, the map amplifies and suppresses certain landscape regions and contributes to directed movement through a nonlocal spatial kernel. After establishing the well-posedness of the coupled PDE-ODE system, we perform a linear stability analysis to identify critical aggregation strengths. We then perform a rigorous bifurcation analysis to determine the precise solution behavior at a steady state near these critical thresholds, deciding whether the bifurcation is sub- or supercritical and the stability of the emergent branch. Based on our analytical findings, we highlight several interesting biological consequences. First, we observe that whether the spatial map functions as attractive or repulsive depends precisely on the map’s relative excitation rate versus adaptory rate: when the excitatory effect is larger (smaller) than the adaptatory effect, the map is attractive (repulsive). Second, in the absence of growth dynamics, populations can only form a single aggregate. Therefore, the presence of intraspecific competition is necessary to drive multi-peaked aggregations, reflecting higher-frequency spatial patterns. Finally, we show how subcritical bifurcations can trigger abrupt shifts in average population abundance, suggesting a tipping-point phenomenon in which moderate changes in movement parameters can cause a sudden population decline.

Overview

A big motivation for this paper is the idea that movement is not just driven by what is happening right now, but also by a spatial map that is being updated over time. That gives a coupled PDE-ODE system: the PDE describes the population density, while the ODE describes the evolving map. One of the main points of the paper is to show that this system is mathematically well posed, and then to understand when the homogeneous state loses stability and gives rise to patterned steady states through a bifurcation analysis.

From the analysis, a few interesting biological conclusions come out quite clearly. The map can behave either “attractively” or “repulsively” (in the sense that the population profile may be in- or out-of-phase with the spatial map) depending on the balance between uptake and decay of the spatial map, so the same modelling framework can encode very different movement responses. The paper also shows that if population dynamics is removed, the system can only support a single aggregate, which means that intraspecific competition is what drives the appearance of more complicated multi-peaked patterns.

Another interesting part of the story is through the bifurcation structure: subcritical bifurcations can produce abrupt changes in average population abundance, which gives a possible tipping-point mechanism: moderate parameter changes in the movement dynamics can lead to a sudden drop in population size.

Keywords

reaction-diffusion-aggregation system; coupled nonlocal PDE-ODE system; well-posedness; bifurcation; stability; patterned solutions

MSC 2020

35B32; 35K57; 35B35; 35B36; 92D25

Links

• DOI / Journal page
• Preprint (arXiv)