Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations

• Published in: Journal of mathematical biology Vol. 70; no. 4; pp. 709 - 743
• Main Authors: Madzvamuse, Anotida, Ndakwo, Hussaini S., Barreira, Raquel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2015; Springer Nature B.V
• Subjects: Applications of Mathematics; Biological Transport, Active; Computer Simulation; Diffusion; Finite Element Analysis; Kinetics; Linear Models; Mathematical and Computational Biology; Mathematical Concepts; Mathematics; Mathematics and Statistics; Models, Biological; Finite element method; Pattern formation; Cross-diffusion reaction systems; Parameter space identification; Planary domains; 92Bxx; Cross-diffusion driven instability; 37D99; 65M60; 35K57
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-014-0779-6

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 u and v 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.