Saddle–node-Hopf bifurcation of a delayed host-generalist parasitoid system with nonlocal competition and host-taxis

The host-taxis mechanism with time delay and nonlocal competition are innovatively introduced into a host-generalist parasitoid system for studying pine wilt disease control. First, the well-posedness of the model is established. The existence and stability of the equilibrium points are demonstrated. The existence of Turing, Hopf, and Turing–Hopf bifurcations is analyzed. Subsequently, the normal form of the Turing–Hopf bifurcation is derived using the multiple time scales method. Furthermore, the normal form is linearly transformed into a computationally tractable equivalent form. Second, this study performs numerical simulations from two aspects. On the one hand, with time delay serving as the Hopf bifurcation parameter, it is found that when the delay exceeds its critical value, the system transitions from a homogeneous steady state to an inhomogeneous periodic solution. Research on delays shows that shorter search delays aid epidemic control. Moreover, with fixed delay, parasitoids with weak host-taxis are better for prevention. On the other hand, to investigate the combined effects of host-taxis and time delay, the system undergoing a saddle–node-Hopf bifurcation is simulated. Rich dynamical phenomena, such as various inhomogeneous steady states and inhomogeneous periodic solutions, are obtained. Research on the combined effects of host-taxis and time delays shows that purposefully adjusting these two parameters can steer the population toward a stable state, facilitating the control of pine wilt disease.