Global well-posedness and spatially inhomogeneous Hopf bifurcation in a predator-prey system with indirect predator-taxis

This paper explores a predator-prey system featuring indirect predator-taxis, where prey exhibit a repellent response triggered by chemicals secreted by predator. We first establish the global existence and uniform boundedness of classical solutions for the system in any spatial dimension, assuming that the functional response $g(u,v)$ is bounded. Additionally, under the assumption of quadratic decay in the prey population density, we prove the global existence and uniform boundedness of classical solutions for the system in up to two spatial dimensions, assuming that the functional response $g(u,v)$ is sublinear. Linear stability analysis reveals that indirect predator-taxis plays a crucial role in pattern formation. For the Lotka-Volterra type functional response, we demonstrate global stability of the positive constant steady state by constructing an appropriate Lyapunov functional. Conversely, for the Beddington-DeAngelis functional response, we investigate Hopf bifurcation in the predator-prey system with indirect predator-taxis. To compute the normal form of this bifurcation, we introduce an efficient new algorithm treating the taxis coefficient as a perturbation parameter. Using this algorithm, we analyze the direction and stability of taxis coefficient-induced Hopf bifurcation. Finally, numerical simulations are conducted to validate our analytical findings.