| Abstract: |
| Nonlocal aggregation-diffusion models coupled with spatial maps provide a framework to capture cognitive and memory-driven effects in animal movement and population dynamics. In this study, we investigate a one-dimensional reaction-diffusion-aggregation system in which population dynamics are coupled with a dynamically evolving map that modulates movement via nonlocal interactions. We first establish the well-posedness of the coupled PDE-ODE system, and then conduct linear stability analysis to identify critical aggregation parameters. A rigorous bifurcation analysis is performed to characterize steady-state behavior near critical thresholds, including the type of bifurcation and the stability of emerging branches. Our results reveal several key biological insights. The spatial map acts as attractive or repulsive depending on the balance between excitation and adaptation effects. In the absence of growth, only single-peak aggregation occurs, indicating that intraspecific competition is essential for multi-peak pattern formation. Moreover, subcritical bifurcations can induce abrupt shifts in population levels, suggesting tipping-point dynamics under moderate parameter changes. |
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