Pattern formation in a nonlinear model for animal coats
The reaction-diffusion system of the form \[ u_t=\varepsilon \Delta u + f(u,v), \quad v_t=d\varepsilon \Delta v + g(u,v) \] is considered; the nonlinearities \(f, g\) satisfy the conditions for the occurence of a Turing instability, e.g. it may be a model of the formation of the animal coat patterns due to Murray: \[ f(u,v)=a-u-h(u,v), \quad g(u,v)=\alpha (b-v) -h(u,v), \] with \(h(u,v)=(1+u+Ku^2)^{-1}\rho uv\). The early stages of the pattern formation process for large domain sizes are described. It is shown that the same mechanism which is responsible for spinodal decomposition in the Cahn-Hilliard equation is responsible for the pattern formation. The authors use techniques of dynamical systems, viewing solutions in terms of nonlinear semiflows.
scientific article; zbMATH DE number 1915670
scientific article