title: Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations.
abstract: By introducing linear cross-diffusion for a two-component reaction-diffusion system with activator-depleted reaction kinetics (Gierer and Meinhardt, Kybernetik 12:30-39, 1972; Prigogine and Lefever, J Chem Phys 48:1695-1700, 1968; Schnakenberg, J Theor Biol 81:389-400, 1979), we derive cross-diffusion-driven instability conditions and show that they are a generalisation of the classical diffusion-driven instability conditions in the absence of cross-diffusion. Our most revealing result is 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 the other. Furthermore, it is no longer necessary to have an activator-inhibitor mechanism as premises for pattern formation, activator-activator, inhibitor-inhibitor reaction kinetics as well as short-range inhibition and long-range activation all have the potential of giving rise to cross-diffusion-driven instability. To support our theoretical findings, we compute cross-diffusion induced parameter spaces and demonstrate similarities 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 parameter spaces. Our numerical experiments validate our theoretical predictions that parameter spaces induced by cross-diffusion in both the [Formula: see text] and [Formula: see text] components of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spaces of the cross-diffusion induced parameter spaces. Our results allow experimentalists to have a wider range of parameter spaces from which to select reaction kinetic parameter values that will give rise to spatial patterning in the presence of cross-diffusion.
authors.affiliation: Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Pevensey III, 5C15, Brighton, BN1 9QH, UK, a.madzvamuse@sussex.ac.uk.
authors.forename: Anotida
authors.initials: A
authors.lastname: Madzvamuse
authors.forename: Hussaini S
authors.initials: HS
authors.lastname: Ndakwo
authors.forename: Raquel
authors.initials: R
authors.lastname: Barreira
mesh_terms: D001693:Biological Transport, Active; D003198:Computer Simulation; D004058:Diffusion; D020342:Finite Element Analysis; D007700:Kinetics; D016014:Linear Models; D055641:Mathematical Concepts; D008954:Models, Biological