Stability and bifurcation of a predator–prey model with variable search rate in advective environments

In this paper, we consider a predator–prey model with variable search rate in advective environments. First, by applying the variational expression of the principal eigenvalue, we analyze its continuity and monotonicity with respect to parameters, and combine techniques such as priori estimates and comparison principle to obtain the global asymptotic stability of trivial and semi-trivial steady-state solutions. The complete classification of long time dynamic behaviors of the system is conducted using advection rate $q$ and half-saturation constant $g$ which is used to reflect the magnitude of search rate as parameters. Moreover, owing to the properties of the principal eigenvalue and principal eigenfunction, we establish the local existence and stability of the positive steady state, as well as the global existence, through bifurcation theory and persistence theory, respectively. Furthermore, for such non-monotone system with advective term and nonlinear functional response, we prove that it exhibits a unique spatially inhomogeneous positive steady state for a small advection rate when the predator has an intermediate search rate. More interestingly, we observe complex phenomena, including spatially inhomogeneous periodic solution, and find that both prey and predator tend to move towards the middle of the river when the search rate is large.