On the robustness of discontinuous patterns in degenerate reaction–diffusion systems with perturbed hysteresis

Abstract

In this paper, we study the dynamics of a morphogenesis model which reproduces the formation of discontinuous patterns observed in numerous biological or ecological systems. This model is determined by a degenerate reaction–diffusion system with hysteresis in the non-diffusive equation. We focus on the robustness of discontinuous patterns under the action of a perturbation of the hysteresis process. We analyze the bifurcations of homogeneous stationary solutions to this nonlinear model and prove that the trivial solution is the only one to resist to a perturbation of strong intensity. We then prove an original result on the structural transformation of discontinuous patterns, which are seen to react by acquiring a supplementary discontinuity jump under the effect of a perturbation of small intensity. We interpret the supplementary jump for ecological systems as the possible emergence of an invasive ecosystem. Numerical simulations and animations are finally provided to guide intuition on the complex morphogenesis process of this dynamical system.