Diffusive spatial movement with memory and carrying capacity delays under periodic boundary conditions

In this paper, we introduce a reaction–diffusion equation incorporating memory and carrying capacity delays. We analyze the influence of the memory-based diffusion coefficient on the stability of positive constant equilibrium of the model. When the memory-based diffusion is small, the model can undergo Turing bifurcation. When the memory-based diffusion is big, the model exhibits richer dynamical behavior: (1) For the model with spatial memory delay τ1 only, the positive constant equilibrium is unstable for all τ1>0 and model can undergo Hopf bifurcation at the critical value. (2) For the model with carrying capacity delay τ2 only, we distinguish two cases based on the magnitude of β. For big β, the model can undergo Hopf bifurcation at the critical values. (3) For τ1,τ2>0, spatiotemporal patterns exhibit richer dynamical regimes. Using the geometric approach proposed by Gu et al., (2005), we find richer dynamical behavior. Finally, we validate the conclusions through numerical simulations.