Spatiotemporal pattern formations in a two-layer coupled reaction-diffusion Lengyel-Epstein system

Abstract

Spatiotemporal pattern formations in two-layer coupled reaction-diffusion Lengyel-Epstein system with distributed delayed couplings are investigated. Firstly, for the original decoupled system, it is proved that when the intra-reactor diffusion rate $\ep$ of the inhibitor is sufficiently small and the intra-reactor diffusion rate $d$ of the inhibitor is large enough, then the subsystem can exhibit non-constant positive steady state $(\widetilde{u}(x;\ep),\widetilde{v}(x;\epsilon))$ with large amplitude, and that as the parameter $\tau$ varies, the stability of $(\widetilde{u}(x;\ep),\widetilde{v}(x;\epsilon))$ changes, leading to the emergence of periodic solutions via Hopf bifurcation. Secondly, for the two-layer coupled system, the stability of the symmetric steady state $(\widetilde{u}(x;\ep),\widetilde{v}(x;\epsilon),\widetilde{u}(x;\ep),\widetilde{v}(x;\epsilon))$ is studied by treating $k_1,k_2$ (the inter-reactor diffusion rates) and $\al$ (the delay parameter) as the main parameters. In case of non-delayed couplings, the first quadrant of the $(k_1, k_2)$ parameter space can be divided into two regions: one is stable region, the other one is unstable region, and the two regions have the common boundary, which is the primary Turing bifurcation curve. In case of delayed couplings, it is shown that the first quadrant of the $(k_1, k_2)$ parameter space can be re-divided into three regions: the first one is unstable region, the second one is stable region, while the third one is the potential \lq\lq bifurcation\rq\rq region, where Hopf bifurcation may occur for suitable $\al$. Our analysis is mainly based on the singular perturbation techniques and the implicit function theorem, and the results show some different phenomena from those of the original decoupled system in one reactor.