Dynamics of a coupled nonlocal PDE-ODE system with spatial memory: well-posedness, stability, and bifurcation analysis
Yurij Salmaniw, Di Liu, Junping Shi, Hao Wang
TL;DR
This work analyzes a one-dimensional reaction–diffusion–aggregation model coupled to a dynamically evolving spatial memory map $k(x,t)$, where $k_t = g_1(u) - g_2(u) k$ and movement is biased by a nonlocal term via $ar{k} = G*k$ with an exponential kernel. By proving well-posedness and establishing a local equivalence to a local parabolic–ODE–elliptic system, the authors perform a linear stability and Crandall–Rabinowitz bifurcation analysis to characterize when nonuniform steady states arise and whether the bifurcations are subcritical or supercritical. A central finding is that the map can act attractively or repulsively depending on the relative excitation and adaptation rates near the homogeneous steady state, which, together with population growth, drives multi-peak patterns and potential tipping-point behavior. Numerically, the framework reveals how growth, memory encoding, and kernel range interact to produce in-phase or out-of-phase patterns and to determine the stability and direction of bifurcating branches, offering insights into ecological pattern formation and abrupt abundance shifts.
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.