A predator–prey model with taxis mechanisms and stage structure for the predator**This work was supported by NSFC Grant 11771110.

Abstract

We study the initial-boundary value problem of a predator–prey model with two taxis strategies and stage structure for the predator: where is a smooth bounded domain, constants χ, d1, d2, d3, a, b, c, k, r are supposed to be positive, while ρ is nonnegative. For n = 1 with ρ ⩾ 0 and n = 2 with ρ = 0, the global existence and boundedness of classical solution are established. The boundedness results clarify how the taxis-type mechanisms affect the upper bounds of the solution. The linearized stabilities of the positive constant steady state and predator-free steady state are investigated secondly. It is found that, for certain parameters, the large value of ρ may result in the instability of the positive constant steady state while the smaller ρ stabilizes it. Outside these parameter regimes, when χ is large or ρ is large, the positive constant steady state is unstable. We then show bifurcations of steady states and time-periodic solutions by using ρ as the bifurcation parameter via local bifurcation and Hopf bifurcation theories. It is shown that, when χ is small enough, the steady state bifurcation can be expected. Whereas, the Hopf bifurcation may happen for any value of χ. Extensive numerical simulations are preformed to illustrate the emergence of steady state patterns and time-periodic patterns. Moreover, by constructing Lyapunov functional, the global stability of the predator-free steady state is established.