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