Patterns formation in hyperbolic reaction-diffusion models with cross-diffusion

TL;DR

This paper investigates pattern formation in hyperbolic reaction-diffusion models with cross-diffusion, analyzing stability, bifurcations, and transient regimes, with applications to the hyperbolic Schnakenberg model.

Contribution

It introduces a hyperbolic reaction-diffusion framework with cross-diffusion, analyzing pattern formation and stability using linear and weakly nonlinear methods.

Findings

Identification of Hopf, Turing, and Wave bifurcations in the model.

Analytical solutions for pattern amplitude evolution.

Numerical validation of theoretical predictions.

Abstract

A class of hyperbolic reaction--diffusion models with cross-diffusion is derived within the context of Extended Thermodynamics. Linear stability analysis is performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. Emphasis is given to the occurrence of Hopf, Turing and Wave bifurcations. The weakly nonlinear analysis is then employed to deduce the equation governing the time evolution of pattern amplitude and to obtain the analytical approximated solution. The influence of the hyperbolic structure of the model on the pattern formation as well as on the transient regimes is highlighted. The theoretical predictions are illustrated on the hyperbolic Schnakenberg model and linear and weakly nonlinear stability are investigated both analytically and numerically.