Spatiotemporal evolution in a (2+1)-dimensional chemotaxis model

TL;DR

This paper investigates the complex spatiotemporal behaviors of a (2+1)-dimensional chemotaxis model, revealing conditions for steady states, divergence, and chaotic pattern formation through simulations and wavelet analysis.

Contribution

It provides new insights into the nonlinear dynamics and pattern formation in a (2+1)-dimensional Keller-Segel chemotaxis model with logistic growth.

Findings

Steady states occur at high chemotactic coefficients or low growth rates.

Solutions diverge in finite time when growth rate exceeds a critical value.

Chaotic patterns are observed in certain parameter regimes.

Abstract

Simulations are performed to investigate the nonlinear dynamics of a (2+1)-dimensional chemotaxis model of Keller-Segel (KS) type with a logistic growth term. Because of its ability to display auto-aggregation, the KS model has been widely used to simulate self-organization in many biological systems. We show that the corresponding dynamics may lead to a steady-state, divergence in a finite time as well as the formation of spatiotemporal irregular patterns. The latter, in particular, appear to be chaotic in part of the range of bounded solutions, as demonstrated by the analysis of wavelet power spectra. Steady states are achieved with sufficiently large values of the chemotactic coefficient $(\chi)$ and/or with growth rates $r$ below a critical value $r_c$. For $r > r_c$, the solutions of the differential equations of the model diverge in a finite time. We also report on the pattern formation regime for different values of $\chi$, $r$ and the diffusion coefficient $D$.