In this work we are interested in describing the mechanism of pattern formation for a reaction-diffusion system with nonlineardiffusion terms (which take into account the self and the cross-diffusion effects). The reaction terms are chosen of theLotka-Volterra type in the competitive interaction case. The cross-diffusion is proved to be the key mechanism of patternformation via a linear stability analysis.A weakly nonlinear multiple scales analysis is carried out to predict the amplitude and the form of the pattern close to thebifurcation threshold. In particular, the Stuart-Landau equation which rules the evolution of the amplitude of the most unstable mode is found. In the subcritical case the solutions predicted by the weakly nonlinear analysis are compared with the numerical solutions of the original system. Close to the threshold they show a good agreement. With the increasing distance from the bifurcation value of the cross-diffusion parameter, the weakly nonlinear analysis fails and a Fourier--Galerkin approach isadopted. A system of ODEs is then derived which captures the behavior of the system far from bifurcation.On the other hand, in order to correctly describe the amplitude of the pattern in the subcritical case, the quintic Stuart-Landauequation has to be derived.The bifurcation diagram shows a range of the bifurcation parameter in which two qualitatively different stable states coexist (the origin and two large amplitude branches). The existence of different stable states for one single value of the parameterallows for the possibility of hysteresis. The evolution of the pattern corresponding to the hysteresis cycle is shown.Moreover, a good agreement is obtained between the numerical solution of the reaction-diffusion system and the weakly nonlinear solution to the fourth order. Finally, we also perform a weakly nonlinear perturbation analysis in the case of a 2-d spatial domain and we find when bifurcation occurs via a simple eigenvalue patterns such rolls, square or rhombi can be supported by the model equation. Otherwise, if bifurcation occurs via a double eigenvalue, more complex patterns are also possible, for example hexagonal patterns.
|Stato di pubblicazione||Published - 2009|