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

journal contribution
posted on 2023-06-08, 18:09 authored by Anotida MadzvamuseAnotida Madzvamuse, Hussaini S Ndakwo, Raquel Barreira
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 presented.) and (Formula presented.) 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.

History

Publication status

  • Published

File Version

  • Published version

Journal

Journal of Mathematical Biology

ISSN

0303-6812

Publisher

Springer Verlag

Issue

4

Volume

70

Page range

709-743

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2014-09-01

First Compliant Deposit (FCD) Date

2014-09-01

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