Spatiotemporal dynamics of the diffusive mussel-algae model near Turing-Hopf bifurcation. (English) Zbl 1382.35035
Summary: Intertidal mussels can self-organize into periodic spot, stripe, labyrinth, and gap patterns ranging from centimeter to meter scales. The leading mathematical explanations for these phenomena are the reaction-diffusion-advection model and the phase separation model. This paper continues the series studies on analytically understanding the existence of pattern solutions in the reaction-diffusion mussel-algae model. The stability of the positive constant steady state and the existence of Hopf and steady-state bifurcations are studied by analyzing the corresponding characteristic equation. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the explicit dynamical classification in its neighborhood by calculating and investigating the normal form on the center manifold. Using theoretical and numerical simulations, we demonstrates that this TH interaction would significantly enhance the diversity of spatial patterns and trigger the alternative paths for the pattern development.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 92D40 | Ecology |
| 35B36 | Pattern formations in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
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