Journal
JOURNAL OF MATHEMATICAL BIOLOGY
Volume 70, Issue 4, Pages 709-743Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00285-014-0779-6
Keywords
Cross-diffusion reaction systems; Cross-diffusion driven instability; Parameter space identification; Pattern formation; Planary domains; Finite element method
Categories
Funding
- EPSRC [EP/J016780/1]
- EPSRC [EP/H020349/1, EP/J016780/1] Funding Source: UKRI
- Engineering and Physical Sciences Research Council [EP/H020349/1, EP/J016780/1] Funding Source: researchfish
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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 and 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.
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