Roles of delay on a food-limited predator–prey model with prey-taxis

Due to the attraction or repulsion towards some chemical signals, prey and predator may move in a specific direction. We establish a food-limited predator–prey model with delay and prey-taxis under the Neumann boundary conditions. Without diffusion and delay, for certain ranges of the capturing rate and the half-saturation constant, the positive equilibrium in the food-limited environment could change its stability as the intrinsic growth rate of predator varies from large to small, whereas the stability of the positive equilibrium is independent in the intrinsic growth rate of predator in the logistic growth case. Prey-taxis can stabilise diffusion-driven instabilities and influence the biomass of predator and prey. The joint effect of delay and diffusion can derive spatially nonhomogeneous periodic patterns by spatially nonhomogeneous Hopf bifurcations. Moreover, when delay exceeds the critical threshold, the amplitude of the periodic solution increases, which means that the risk of population extinction also increases, as delay increases. It is worth noting that to some extent, the food-limited environment could prevent the occurrence of periodic solutions induced by delay.