UBC Math Bio Seminar: Anotida Madzvamuse

In this talk, I will present a roadmap for deriving minimal necessary conditions for diffusion-driven instability for a 3-component reaction-diffusion system with linear cross-diffusion. For the reaction kinetics, we postulate and formulate a new 3-component phenomenological molecular interaction network exhibiting one activator and two inhibitor molecular species. The approach for establishing these necessary conditions is divided into two parts. For the first part, we consider a 3-component reaction kinetics where spatial variations are neglected, and establish the necessary conditions in order for the unique uniform steady state of the system to be linearly asymptotically stable. These conditions are based on the Routh-Hurwitz criterion for a cubic polynomial. The second part, seeks to establish necessary conditions such that when introducing spatial variations, the uniform steady state becomes linearly unstable. The theoretical instability conditions in the presence of spatial variations emerge from the violation of at least one of the Routh-Hurwitz conditions for stability. It turns out that the mathematical derivation of these necessary conditions results in various structured polynomials in terms of the wave numbers. By exploiting these structures, we are able to derive necessary conditions for diffusion-driven instability, which are a substantial generalisation of the conditions for Turing diffusion-driven instability for a two-component system. To verify the validity of these conditions, we systematically derive diffusion-driven instability parameter spaces by picking the full cross-diffusion tensor matrices from real physical applications. To support theoretical findings, we then select parameter values from the obtained parameter spaces and by using the finite element method in 2-space dimensions, we illustrate the validity of our findings by generating cross-diffusion-driven patterns on a circular domain, reminiscent of a petri dish.