The role of predator evasion mediated by chemical signaling is studied in a diffusive prey–predator model when prey-taxis is taken into account (model A) or not (model B) with taxis strength coefficients χ and ξ, respectively. In the kinetic part of the models, it is assumed that the rate of prey consumption includes functional responses of Holling, Beddington–DeAngelis or Crowley–Martin. Existence of global-in-time classical solutions to model A is proved in space dimension n = 1 while to model B for any n ≥ 1. The Crowley–Martin response combined with bounded rate of signal production precludes blow-up of solution in model A for n ≤ 3. Local and global stability of a constant coexistence steady state which is stable for the corresponding ordinary differential equation (ODE) and purely diffusive model are studied along with mechanism of Hopf bifurcation for model B when χ exceeds some critical value. In model A, it is shown that prey-taxis may destabilize the coexistence steady state provided χ and ξ are big enough. Numerical simulation depicts emergence of complex space-time patterns for both models and indicates existence of solutions to model A which blow-up in finite time for n = 2.

中文翻译：

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具有扩散和猎物趋向性的捕食者 - 猎物模型中的排斥趋化性和捕食者逃避

当猎物出租车被考虑（模型A）或不考虑（模型B）与出租车强度系数时，由化学信号介导的捕食者逃避的作用在扩散的猎物 - 捕食者模型中进行了研究χ和ξ， 分别。在模型的动力学部分，假设猎物消耗率包括 Holling、Beddington-DeAngelis 或 Crowley-Martin 的功能反应。在空间维度上证明了模型 A 的全局时间经典解的存在性n = 1同时为任何建模 Bn ≥ 1. Crowley-Martin 响应与有限的信号产生速率相结合，防止了模型 A 中的解爆n ≤ 3. 研究了对应常微分方程（ODE）和纯扩散模型稳定的常数共存稳态的局部和全局稳定性以及模型B的Hopf分岔机制χ超过某个临界值。在模型 A 中，表明猎物出租车可能会破坏提供的共存稳定状态χ和ξ足够大。数值模拟描述了两种模型的复杂时空模式的出现，并表明模型 A 的解的存在，该解在有限时间内爆炸n = 2.