Existence and Turing instability of positive solutions for a predator–pest model with additional food

To better explore the dynamics of pests, this paper deals with a brand-new predator–pest model with diffusion and additional food. The existence and diffusion-driven Turing instability of positive constant solutions are discussed. We obtain that for additional food of a certain quality and quantity, there exists a critical value such that the model can produce four forms of positive constant solutions as the predation rate of predators is greater than the critical value, and only one form of positive constant solution as the predation rate is less than the critical value. For predator–pest model, which is a new finding indeed. Meanwhile, we conclude that the introduction of diffusion can lead to Turing instability of positive constant solutions. This indicates that the model is likely to produce spatial pattern with certain conditions.