Turing pattern analysis of a delayed reaction–diffusion eco-epidemiological disease model with fatal disease in the prey

In order to study the effects of fear effect and time delay on the transmission of ecological epidemics, a delayed response eco-epidemic model in which the prey has a fatal disease has been proposed. Initially, the existence and boundedness of system solutions are studied. Subsequently, the existence of non-negative equilibria is obtained and their local dynamic behavior is analyzed. In addition, the global dynamic of the system boundary equilibrium is analyzed, and the sufficient conditions for its asymptotic stability are obtained. The conditions of turing instability are obtained by further study of the diffusion equation. Furthermore, taking delay as a branch parameter, the condition of Hopf bifurcation is discussed. Finally, the theoretical results are verified by numerical simulation. The results show that the higher the population density of infected prey in the same area, the lower the population density of susceptible prey and predators and the density change trend of both is consistent since the food sources used by predators for their own reproduction are all susceptible prey.