Distinguishing between long-transient and asymptotic states in a biological aggregation model

TL;DR

This paper introduces a new energy minimization technique to distinguish long transient states from true steady states in a biological aggregation model, revealing most multi-peaked solutions are transient.

Contribution

The authors develop a novel approximation and energy analysis method to identify whether aggregation solutions are transient or steady in a biological model.

Findings

Most multi-peaked solutions are long transients rather than steady states.

The new technique accurately predicts the nature of aggregation states.

Long transients can persist for arbitrarily long durations depending on parameters.

Abstract

Aggregations are emergent features common to many biological systems. Mathematical models to understand their emergence are consequently widespread, with the aggregation-diffusion equation being a prime example. Here we study the aggregation-diffusion equation with linear diffusion. This equation is known to support solutions that involve both single and multiple aggregations. However, numerical evidence suggests that the latter, which we term `multi-peaked solutions' may often be long-transient solutions rather than asymptotic steady states. We develop a novel technique for distinguishing between long transients and asymptotic steady states via an energy minimisation approach. The technique involves first approximating our study equation using a limiting process and a moment closure procedure. We then analyse local minimum energy states of this approximate system, hypothesising that these will correspond to asymptotic patterns in the aggregation-diffusion equation. Finally, we verify our hypotheses through numerical investigation, showing that our approximate analytic technique gives good predictions as to whether a state is asymptotic or a long transient. Overall, we find that almost all twin-peaked, and by extension multi-peaked, solutions are transient, except for some very special cases. We demonstrate numerically that these transients can be arbitrarily long-lived, depending on the parameters of the system.