Percorrer por autor "Ndakwo, Hussaini"
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- Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulationsPublication . Madzvamuse, Anotida; Ndakwo, Hussaini; Barreira, RaquelBy introducing linear cross-diffusion for a two-component reaction-diffusion system withactivator-depletedreaction kinetics (Gierer and Meinhardt,Kybernetik 12:30–39,1972;PrigogineandLefever,JChemPhys48:1695–1700,1968;Schnakenberg, J Theor Biol 81:389–400,1979), we derivecross-diffusion-driveninstability conditions and show that they are a generalisation of the classical diffusion-driveninstabilityconditionsintheabsenceofcross-diffusion.Ourmostrevealingresultis that, in contrast to the classical reaction-diffusion systems without cross-diffusion,it is no longer necessary to enforce that one of the species diffuse much faster than theother.Furthermore,it is no longer necessary to have an activator–inhibitor mecha-nism as premises for pattern formation, activator–activator,inhibitor–inhibitorreac-tion kinetics as well asshort-range inhibitionandlong-range activationall have thepotential of giving rise to cross-diffusion-driven instability. To support our theoreti-cal findings, we compute cross-diffusion induced parameter spaces and demonstratesimilarities and differences to those obtained using standard reaction-diffusion theory.Finite element numerical simulations on planary square domains are presented to back-up theoretical predictions. For the numerical simulations presented, we choose parameter values from and outside the classical Turing diffusively-driven instability space;outside, these are chosen to belong to cross-diffusively-driven instability parameterspaces. Our numerical experiments validate our theoretical predictions that parameterspaces induced by cross-diffusion in both theuandvcomponents of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spacesof the cross-diffusion induced parameter spaces. Our results allow experimentalists tohave a wider range of parameter spaces from which to select reaction kinetic parametervalues that will give rise to spatial patterning in the presence of cross-diffusion.
- Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusionPublication . Madzvamuse, Anotida; Ndakwo, Hussaini; Barreira, RaquelThis article presents stability analytical results of a two compo-nent reaction-diffusion system with linear cross-diffusion posed on continuouslyevolving domains. First the model system is mapped from a continuously evolv-ing domain to a reference stationary frame resulting in a system of partialdifferential equations with time-dependent coefficients. Second, by employingappropriately asymptotic theory, we derive and prove cross-diffusion-driven in-stability conditions for the model system for the case of slow, isotropic domaingrowth. Our analytical results reveal that unlike the restrictive diffusion-driveninstability conditions on stationary domains, in the presence of cross-diffusioncoupled with domain evolution, it is no longer necessary to enforce cross norpure kinetic conditions. The restriction toactivator-inhibitorkinetics to inducepattern formation on a growing biological system is no longer a requirement.Reaction-cross-diffusion models with equal diffusion coefficients in the principalcomponents as well as those of theshort-range inhibition, long-range activa-tionandactivator-activatorform can generate patterns only in the presence ofcross-diffusion coupled with domain evolution. To confirm our theoretical find-ings, detailed parameter spaces are exhibited for the special cases of isotropicexponential, linear and logistic growth profiles. In support of our theoreticalpredictions, we present evolving or moving finite element solutions exhibitingpatterns generated by ashort-range inhibition, long-range activationreaction-diffusion model with linear cross-diffusion in the presence of domain evolution.