Hopf-Hopf bifurcation and spatially staggered time-periodic orbits induced by the joint effects of predator-taxis and delay

When Hopf bifurcation with different modes interact with each other, making the positive constant steady state unstable through Hopf-Hopf bifurcation, it often means that there will be relatively rich time-periodic orbits. It is meaningful and necessary to study the types and stability of these time-periodic orbits. In this paper, we provide a theoretical tool to reveal the time-periodic patterns caused by Hopf-Hopf bifurcation for partial functional differential equations with nonlinear diffusion by means of the normal form method and center manifold reduction. We derive the explicit formulae for the coefficients of normal form associated with Hopf-Hopf bifurcation, which can be applied for the systems with or without nonlinear diffusion. A delayed predator-prey model with predator-taxis is performed to illustrate our results. It turns out that Hopf-Hopf bifurcation will occur due to the combined effects of predator-taxis and time delay. And we theoretically prove that stable spatially homogeneous time-periodic orbits, stable/bistable time-periodic orbits with spatially staggered oscillations as well as transient quasi-periodic orbits exist near the Hopf-Hopf bifurcation point. These phenomena can not occur for the same system with only time delay or only predator-taxis. Some numerical simulations are also presented to verify our analysis expected on theoretical grounds.