Effect of delay on pattern formation of a Rosenzweig–MacArthur type reaction–diffusion model with spatiotemporal delay

In this paper, a predator–prey reaction–diffusion model with Rosenzweig–MacArthur type functional response and spatiotemporal delay is investigated through using the tool of Turing bifurcation theories. First, by taking the average time delay as a bifurcation parameter, conditions of occurrence of Turing bifurcation are obtained through employing the Routh–Hurwitz criteria. Second, as the average time delay varies the amplitude equations of Turing bifurcation patterns including spots pattern and stripes pattern are also obtained through the multiple scale perturbation method. Finally, the two kinds of spatiotemporal evolution distributions of species such as spots pattern and stripes pattern are shown to illustrate theoretical results.