Time delay in the resource limitation of the prey is incorporated into a diffusive predator-prey model with generalist predator. By analyzing the eigenvalue spectrum, time delay inducing instability and Hopf bifurcation are investigated. Some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are obtained by utilizing the normal form method and center manifold reduction for partial functional differential equation. The results suggest that time delay can destabilize the stability of coexisting equilibrium and induce bifurcating periodic solution when it increases through a certain threshold.

Since the distribution of plankton is always uneven, the nonlocal phytoplankton competition term indicates the spatial weighted mean of phytoplankton density, which is introduced into the plankton model with toxic substances effect to study the corresponding dynamic behavior. The stability of the positive equilibrium point and the existence of Hopf bifurcations are discussed by analysing the distribution of eigenvalues. The direction and stability of bifurcation periodic solution are researched based on an extended central manifold method and normal theory. Finally, spatially inhomogeneous oscillations are observed in the vicinity of the Hopf bifurcations. Through numerical simulation, we can observe that the system without nonlocal competition term only generates homogeneous periodic solution, and inhomogeneous periodic solution will produce only when both diffusion and nonlocal competition exist simultaneously. We can also see that when the toxin-producing rate of each phytoplankton is in an appropriate range, the system with nonlocal competition generates a stability switch with inhomogeneous periodic solution, when the value of time delay is in a certain interval, then Hopf bifurcations will appear, and with the increase of time delay, the quantity of plankton will eventually become stable.