Pattern dynamics in a reaction–diffusion predator–prey model with Allee effect based on network and non-network environments

In this paper, we establish a predator–prey model with Beddington–Deangelis functional response. For this model, firstly, the existence of the positive equilibrium point and the conditions for the Turing instability are studied. Then the amplitude equation is derived through weakly nonlinear analysis, and the relationship between the selection of pattern and the coefficients of the amplitude equation is obtained. At the same time, through a large number of numerical simulations, we verify the accuracy of the theoretical analysis. Actually, we mainly choose to change the values of parameters r and d2 to study the sensitivity of the pattern to them. When the pattern tends to be stable, there could be the pattern of spots, coexistence of spots and stripes or stripes. Even if they are spot pattern, the spot density will also be different due to the selection of parameters. Finally, we simulate and compare that the network structure(mainly BA and WS) has a certain influence on the time required for pattern stabilization and the distribution of node density. The final results show that the growth rate of prey, diffusion coefficients and network structure all play an important role in the formation of Turing pattern.