Self-organized anomalous aggregation of particles performing nonlinear and non-Markovian random walk

TL;DR

This paper introduces a nonlinear, non-Markovian random walk model demonstrating self-organized anomalous aggregation of particles, revealing conditions for collapse of stationary patterns and transient bi-modal aggregation.

Contribution

It presents a novel model showing how nonlinear, non-Markovian dynamics lead to self-organized anomalous aggregation without heavy-tailed waiting times.

Findings

Self-organized anomaly causes collapse of stationary patterns.

Power-law stationary density-dependent survival functions are observed.

Critical conditions for divergence of mean residence time are identified.

Abstract

We present a nonlinear and non-Markovian random walk model for stochastic movement and the spatial aggregation of living organisms that have the ability to sense population density. We take into account social crowding effects for which the dispersal rate is a decreasing function of the population density and residence time. We perform stochastic simulations of random walk and discover the phenomenon of self-organized anomaly (SOA) which leads to a collapse of stationary aggregation pattern. This anomalous regime is self-organized and arises without the need for a heavy tailed waiting time distribution from the inception. Conditions have been found under which the nonlinear random walk evolves into anomalous state when all particles aggregate inside a tiny domain (anomalous aggregation). We obtain power-law stationary density-dependent survival function and define the critical condition for SOA as the divergence of mean residence time. The role of the initial conditions in different SOA scenarios is discussed. We observe phenomenon of transient anomalous bi-modal aggregation.