Turing-Hopf bifurcation in a diffusive mussel-algae model with time-fractional-order derivative

• A diffusive mussel-algae model with time fractional-order derivative is proposed. • The existence of Turing instability, Hopf bifurcation, Turing-Hopf bifurcation are achieved in the presence of time-fractional derivative. • The influence of the time fractional derivative on the spatiotemporal dynamics is investigated. • The obtained mathematical results are tested numerically. In this paper, we consider a time fractional-order derivative for a diffusive mussel–algae model. The existence of pattern formation was the subject of interest of many previous research works in the case of the diffusive mussel–algae model. Examples include the Turing instability, Hopf bifurcation, Turing-Hopf bifurcation, and others. The presence of the time–fractional–order derivative never been investigated in this model. Next to it ecological relevant, it can generate some important patterns. One of these patterns is produced by the presence of the Turing-Hopf bifurcation. Therefore, our main interest is to analyze the effect of the time fractional–order derivative on the spatiotemporal behavior of the solution, which never been achieved for the mussel-algae model. Besides, Turing–Hopf was studied exclusively on the classical reaction-diffusion systems, where it was also considered for the diffusive mussel-algae model. Thus, our paper puts the fist steps on proving the existence of this type of codimension bifurcation on the diffusive systems with time fractional–order–derivative systems. Further, a suitable numerical simulations are used for confirming the theoretical obtained results.