Turing diffusion-driven instability analysis of a three-component activator-bi-inhibitor reaction–diffusion system with linear cross-diffusion

Abstract This article presents a systematic approach for deriving the necessary and sufficient conditions for diffusion-driven instability for a three-component reaction–diffusion system (RDS) with linear cross-diffusion and activator-bi-inhibitor reaction kinetics. First, by considering the system in the absence of spatial variations, we derive the necessary conditions for the uniform steady state to be linearly asymptotically stable, based on the Routh–Hurwitz criterion. Second, by introducing spatial variations, we derive 17 different sets of necessary conditions under which the uniform steady state becomes linearly unstable. These conditions emerge from the violation of at least one of the Routh–Hurwitz conditions for stability. The derivation of these conditions reduces to studying one quadratic and two cubic polynomials as functions of the wave numbers. This is in contrast to the case of a two-component RDS, where Turing instability is driven by a single quadratic polynomial. To verify the validity of these conditions, we systematically derive diffusion-driven instability parameter spaces by picking cross-diffusion tensor matrices from real physical systems. Sufficient conditions are established by determining parameter values from the generated parameter spaces such that specific modes are isolated for pattern formation. Finite element simulations in 1–space dimension confirm theoretical predictions of linear stability analysis.