In this paper, we studied a diffusive predator-prey system that incorporated three ecological features under homogeneous Neumann boundary conditions. These features included a cooperative hunting functional response, prey-taxis, and a time delay effect in the predator growth rate, all of which influenced predator-prey interactions. These three features, respectively, indicated that predators cooperated while hunting their prey, that predators tended to move in the direction of an increasing prey density gradient, and that a certain amount of time was required for the conversion of captured prey into predator growth. We first analyzed the occurrence of steady-state bifurcation by examining the role of the prey-taxis rate $\chi$. Furthermore, we examined the impact of the time delay effect on the occurrence of Hopf bifurcation, which involved the emergence of spatially homogeneous or nonhomogeneous periodic solutions, as well as stability switches at a positive constant steady state.

In this paper, we studied a generalized reaction-diffusion system that models predator-prey dynamics, incorporating prey-taxis along with a hunting cooperation effect and a logistic source for the predator population, subject to homogeneous Neumann boundary conditions. This system describes a predator-prey interaction in which the prey, via a prey-taxis mechanism, employ group defense strategies against their predators, while the predators, through their functional response and net growth rate, not only cooperate in hunting these defended prey but also exhibit logistic growth dynamics, thereby ensuring the self-regulation of the predator population. We established that solutions to the time- and space-dependent system exhibiting such ecological characteristics exist globally and remain bounded within a one-dimensional spatial domain. Furthermore, using global bifurcation theory, we proved the existence of nonconstant positive steady states. A key ingredient in this analysis is the derivation of uniform a priori estimates for positive steady-state solutions, which play a crucial role in controlling the global solution branches.