Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth

TL;DR

This study analyzes how cellular growth influences the emergence and stability of time-periodic patterns in a two-species Keller-Segel chemotaxis model, using bifurcation analysis and numerical simulations.

Contribution

It provides a rigorous bifurcation analysis showing cellular growth's role in pattern formation and stability in a chemotaxis model.

Findings

Cellular growth induces and stabilizes oscillating patterns.

Large domains are necessary for stable periodic patterns.

A Lyapunov functional exists without cellular growth.

Abstract

This paper investigates the formation of time--periodic and stable patterns of a two--competing--species Keller--Segel chemotaxis model with a focus on the effect of cellular growth. We carry out rigorous Hopf bifurcation analysis to obtain the bifurcation values, spatial profiles and time period associated with these oscillating patterns. Moreover, the stability of the periodic solutions is investigated and it provides a selection mechanism of stable time--periodic mode which suggests that only large domains support the formation of these periodic patterns. Another main result of this paper reveals that cellular growth is responsible for the emergence and stabilization of the oscillating patterns observed in the $3\times3$ system, while the system admits a Lyapunov functional in the absence of cellular growth. Global existence and boundedness of the system in 2D are proved thanks to this Lyapunov functional. Finally, we provide some numerical simulations to illustrate and support our theoretical findings.