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Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations
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Advisor(s)
Abstract(s)
By 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.
Description
Keywords
Cross-diffusion reaction systems Cross-diffusion driven instability Parameter space identification Pattern formation Planary domains Finite element method
Citation
Madzvamuse A., Ndakwo, H.S. , Barreira, R. (2015) Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations. J. Math. Biol., 70, pp. 709-743. doi: 10.1007/s00285-014-0779-6