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.

Stability and Turing Patterns in a Predator–Prey Model with Hunting Cooperation and Allee Effect in Prey Population

In this paper, we are concerned with a diffusive predator–prey model where the functional response follows the predator cooperation in hunting and the growth of the prey obeys the Allee effect. Firstly, the existence and stability of the positive equilibrium are explicitly determined by the local system parameters. It is shown that the ability of the hunting cooperation can affect the existence of the positive equilibrium and stability, and the intrinsic growth rate of the predator population does not affect the existence of the positive equilibrium, but affects the stability. Then the diffusion-driven Turing instability is investigated and the Turing bifurcation value is obtained, and we conclude that when the ability of the cooperation in hunting is weaker than some critical value, there is no Turing instability. The standard weakly nonlinear analysis method is employed to derive the amplitude equations of the Turing bifurcation, which is used to predict the types of the spatial patterns. And it is found that in the Turing instability region, with the parameter changing from approaching Turing bifurcation value to approaching Hopf bifurcation value, spatial patterns emerge from spot, spot-stripe to stripe. Finally, the numerical simulations are used to support the analytical results.

Turing Instability and Amplitude Equation of Reaction-Diffusion System with Multivariable

In this paper, we investigate pattern dynamics with multivariable by using the method of matrix analysis and obtain a condition under which the system loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation with multivariable. This is an effective tool to investigate multivariate pattern dynamics. The example and simulation used in this paper validate our theoretical results. The method presented is a novel approach to the investigation of specific real systems based on the model developed in this paper.