Spatiotemporal dynamics analysis of a semi-discrete reaction-diffusion Mussel-Algae system with advection

Abstract

The spatiotemporal dynamics of a semi-discrete Mussel-Algae system with advection are investigated in this paper. The stability of equilibrium, the direction of Andronov-Hopf bifurcation and Bautin bifurcation to kinetic system are obtained via linear stability analysis, the Lyapunov coefficients and regularity. In order to analyze the Turing instability, the characteristic equations of the advection operator ∇ is considered. Combined with linear analysis, the critical condition of Turing instability conditions for advection term is obtained. Simulations are performed to illustrate the above theoretical results, such as bifurcation diagram, phase orbits and pattern formations. In addition, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. As a result, the theoretical results for Turing instability of some model with advection term may trigger some significance results for further research.