Delay-driven spatial patterns in a predator-prey model with constant prey harvesting

In this paper we study a predator-prey model with time delay and constant prey harvesting. We investigate the effect of the time delay on the stability of the coexistence equilibrium and demonstrate that time delay can induce spatial patterns. Furthermore, a Hopf bifurcation occurs when the delay increase to a critical value. By applying normal form theory and the center manifold theorem, we develop the explicit formulae that determine the stability and direction of the bifurcating periodic solutions. Finally, by numerical simulations we show how the initial condition affects the types of spatial patterns.