In this paper, a predator–prey model with prey-stage structure and prey-taxis is proposed and studied. Firstly, the local stability of non-negative constant equilibria is analyzed. It is shown that non-negative equilibria have the same stability between ODE system and self-diffusion system, and self-diffusion does not have a destabilization effect. We find that there exists a threshold value ξ0 such that the positive equilibrium point of the model becomes unstable when the prey-taxis rate ξ < ξ0, hence the taxis-driven Turing instability occurs. Furthermore, by applying Crandall–Rabinowitz bifurcation theory, the existence, the stability and instability, and the turning direction of bifurcating steady state are investigated in detail. Finally, numerical simulations are provided to support the mathematical analysis.

中文翻译：

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具有猎物阶段结构和猎物出租车的捕食者-猎物模型的平稳模式

本文提出并研究了一种具有猎物阶段结构和猎物-出租车的捕食者-猎物模型。首先，分析了非负常数平衡的局部稳定性。结果表明，非负平衡在ODE系统和自扩散系统之间具有相同的稳定性，并且自扩散没有失稳作用。我们发现存在一个阈值ξ0使得模型的正平衡点变得不稳定，当猎物趋向率ξ < ξ0，因此发生了由出租车驱动的图灵不稳定性。此外，应用Crandall-Rabinowitz分岔理论，详细研究了分岔稳态的存在性、稳定性和不稳定性以及转向的方向。最后，提供数值模拟来支持数学分析。