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On modelling pattern formation by activator-inhibitor systems

Summary

The formation of spatially patterned structures in biological organisms has been modelled in recent years by various mechanisms, including pairs of reaction-diffusion equations

$$u_t = D_{\text{1}} \nabla ^{\text{2}} u + f(u,v)$$

,

$$v_t = D_{\text{2}} \nabla ^{\text{2}} v + g(u,v)$$

.

Their analysis has been by computer simulation. In some cases, u can be interpreted as an activator and v an inhibitor. The following problem is treated: given a “pattern” u = ϕ(x), v = Ψ(x), find a system which has it as a stable stationary solution (stability is used in various senses in the paper). This inverse problem is shown to have solutions for reasonable ϕ and Ψ. The solutions constructed are of activator-inhibitor type with D 2>D 1.

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Sponsored in part by the U.S. Army under Contract No. DAAG29-75-C-0024 and by the N.S.F. under Grant No. MPS-74-06835-A01.

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Fife, P.C. On modelling pattern formation by activator-inhibitor systems. J. Math. Biol. 4, 353–362 (1977). https://doi.org/10.1007/BF00275083

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  • DOI: https://doi.org/10.1007/BF00275083

Keywords

  • Computer Simulation
  • Stochastic Process
  • Inverse Problem
  • Probability Theory
  • Stationary Solution