Global existence of classical solutions and steady-state bifurcation in a prey-taxis predator-prey system with hunting cooperation and a logistic source for predators

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.