Bifurcation phenomena in a single-species reaction-diffusion model with spatiotemporal delay

Abstract

In this paper we investigate bifurcation phenomena in a single-species reaction-diffusion model with spatiotemporal delay under the conditions of the weak and strong kernel functions. We have found that when the weak kernel function is introduced there is Hopf bifurcation but no Turing bifurcation and wave bifurcation to occur, but when the strong kernel function is introduced there exist Hopf bifurcation and wave bifurcation but no Turing bifurcation to occur. Especially, taking the inverse of the average time delay as a bifurcation parameter, we investigate influences of the time delay on the formation of spatiotemporal patterns through the numerical method. Some spatiotemporal patterns induced by Hopf bifurcation and wave bifurcation are respectively shown to illustrate the mechanism of the complexity of spatiotemporal dynamics.