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, d 1, d 2, d 3, 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.

中文翻译：

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具有滑行机制和捕食者阶段结构的捕食者-猎物模型

我们研究了具有两种出租车策略和捕食者阶段结构的捕食者-猎物模型的初始边界值问题：其中是光滑有界域，常数 χ, d 1, d 2, d 3, a, b, c, k, r 应该是正的，而 ρ 是非负的。对于 n = 1 且 ρ ≥ 0 和 n = 2 且 ρ = 0，经典解的全局存在性和有界性成立。有界结果阐明了出租车类型的机制如何影响解决方案的上限。其次研究了正恒定稳态和无捕食者稳态的线性化稳定性。研究发现，对于某些参数，较大的 ρ 值可能会导致正常数稳态的不稳定，而较小的 ρ 会使其稳定。在这些参数范围之外，当 χ 很大或 ρ 很大时，正常数稳态是不稳定的。然后，我们通过局部分岔和 Hopf 分岔理论使用 ρ 作为分岔参数来显示稳态和时间周期解的分岔。结果表明，当 χ 足够小时，可以预期稳态分岔。然而，对于任何 χ 值都可能发生 Hopf 分叉。进行了广泛的数值模拟，以说明稳态模式和时间周期模式的出现。此外，通过构造Lyapunov泛函，建立了无捕食者稳态的全局稳定性。可以预期稳态分叉。然而，对于任何 χ 值都可能发生 Hopf 分叉。进行了广泛的数值模拟，以说明稳态模式和时间周期模式的出现。此外，通过构造Lyapunov泛函，建立了无捕食者稳态的全局稳定性。可以预期稳态分叉。然而，对于任何 χ 值都可能发生 Hopf 分叉。进行了广泛的数值模拟，以说明稳态模式和时间周期模式的出现。此外，通过构造Lyapunov泛函，建立了无捕食者稳态的全局稳定性。