Global boundedness and spatially inhomogeneous Hopf bifurcation in a delayed predator–prey model with dual taxis

This paper investigates a class of delayed reaction–diffusion predator–prey models incorporating both predator-taxis and prey-taxis under homogeneous Neumann boundary conditions. We establish the global existence and uniqueness of positive classical solutions. Furthermore, we prove the global boundedness of weak solutions using a novel extension of the entropy method. While this technique was originally developed for reaction–diffusion models with cross-diffusion, the presence of time delay presents a significant challenge to its direct application. To overcome this, we derive key entropy inequalities essential for obtaining gradient estimates and establishing boundedness. Additionally, we analyze the occurrence of spatially inhomogeneous Hopf bifurcation. By employing the center manifold theorem and normal form theory, we establish a method to compute the associated normal form. Numerical simulations validate both our computational methodology and the theoretical results.