Dynamics of a diffusive competition model with memory effect and spatial heterogeneity

In this paper, a diffusive competition model is formulated by considering the memory-based diffusions of species in their movement and spatial heterogeneity of the resource. The global existence of solution for the proposed model is proved. Under weak competition, the existence and stability of the positive steady state are obtained, when the random diffusion rates are large enough, by using implicit function theorem and some priori estimations. However, when memory-based diffusions dominate the dynamics, spatial-temporal patterns can be observed. We also consider the case that both competition coefficients equal to one, showing that the stabilities of semi-trivial steady states, existence of positive steady state by the method of upper and lower solutions, and bistable dynamics of the model. The results reveal that the memory-based diffusion plays an important role on the competition outcomes. Specifically, the species with slower random diffusion may not help itself to wipe out its competitor because of disadvantageous memory-based diffusion, while the species with faster random diffusion but with advantageous memory-based diffusion may win the competition.