Rich dynamics of a predator–prey model with spatial motion

One of the central issues in ecosystem is the influence of space on the stability of the populations. As a result, a predator–prey model with spatial diffusion and cubic growth rate is investigated. Asymptotical properties of solutions of such spatial model are obtained based on comparison principle. What is more, we show the global stability of positive equilibrium by constructing Lyapunov function. We also show how predator diffuses has impact on the pattern dynamics based on numerical simulations. The results may well enrich the findings in predator–prey systems with spatial motions.