Turing and Turing-Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection

In this paper, we study the stability and the bifurcation properties of the positive interior equilibrium for a reaction-diffusion equation with nonlocal advection. Under rather general assumption on the nonlocal kernel, we first study the local well posedness of the problem in suitable fractional spaces and we obtain stability results for the homogeneous steady state. As a special case, we obtain that "standard" kernels such as Gaussian, Cauchy, Laplace and triangle, will lead to stability. Next we specify the model with a given step function kernel and investigate two types of bifurcations, namely Turing bifurcation and Turing-Hopf bifurcation. In fact, we prove that a single scalar equation may display these two types of bifurcations with the dominant wave number as large as we want. Moreover, similar instabilities can also be observed by using a bimodal kernel. The resulting complex spatiotemporal dynamics are illustrated by numerical simulations.