Spatiotemporal Patterns of a Host-Generalist Parasitoid Reaction-Diffusion Model

In this paper, we study a delayed host-generalist parasitoid diffusion model subject to homogeneous Dirichlet boundary conditions, where generalist parasitoids are introduced to control the invasion of the hosts. We construct an explicit expression of positive steady-state solution using the implicit function theorem and prove its linear stability. Moreover, by applying feedback time delay t as the bifurcation parameter, spatially inhomogeneous Hopf bifurcation near the positive steady-state solution is proved when t is varied through a sequence of critical values. This finding implies that feedback time delay can induce spatially inhomogeneous periodic oscillatory patterns. The direction of spatially inhomogeneous Hopf bifurcation is forward when parameter m is sufficiently large. We present numerical simulations and solutions to further illustrate our main theoretical results. Numerical simulations show that the period and amplitude of the inhomogeneous periodic solution increase with increasing feedback time delay. Our theoretical analysis results only hold for parameter k when it is sufficiently close to 1, whereas numerical simulations suggest that spatially inhomogeneous Hopf bifurcation still occurs when k is larger than 1 but not sufficiently close to 1.