Stable Steady-State Solutions of Some Biological Aggregation Models

Aggregation phenomena occur across the biological sciences, from cell adhesion to insect swarms, animal home ranges to human cities. Understanding the mechanisms by which they may spontaneously emerge has therefore generated much interest from applied mathematicians. Partial differential equations (PDEs) with nonlocal advection offer a popular formalism for studying aggregations. However, the inherent nonlocality, often necessary for ensuring continuum models are well-posed, makes their study technically challenging. Here, we take a different approach by studying a discrete-space system that can be formally related to classical nonlocal PDE approaches via a limiting procedure. We show how to find expressions for the asymptotically stable steady-states of this discrete-space system via an energy functional approach. This allows us to predict the size of aggregations as a function of the underlying movement mechanisms of individual organisms. We apply this to a recent model of cell adhesion, revealing a hysteresis property whereby the existing aggregations may persist even as the adhesion tendency decreases past the bifurcation point. We compare this to numerical solutions of the associated nonlocal PDE system, showing that the hysteresis property predicted by the discrete-space expressions is also present in the continuum system.

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