This paper concerns spatio-temporal pattern formation in a model for two competing prey populations with a common predator population whose movement is biased by direct prey-taxis mechanisms. By pattern formation, we mean the existence of stable, positive non-constant equilibrium states, or nontrivial stable time-periodic states. The taxis can be either repulsive or attractive and the population interaction dynamics is fairly general. Both types of pattern formation arise as one-parameter bifurcating solution branches from an unstable constant stationary state. In the absence of our taxis mechanism, the coexistence positive steady state, under suitable conditions, is locally asymptotically stable. In the presence of a sufficiently strong repulsive prey defense, pattern formation will develop. However, in the attractive taxis case, the attraction needs to be sufficiently weak for pattern formation to develop. Our method is an application of the Crandall–Rabinowitz and the Hopf bifurcation theories. We establish the existence of both types of branches and develop expressions for determining their stability.

中文翻译：

---

带PREY-TAXIS的一捕食者二猎物模型的分岔分析

本文关注两个相互竞争的猎物种群模型中的时空模式形成，该种群具有一个共同的捕食者种群，其运动受到直接猎物出租车机制的影响。通过模式形成，我们指的是存在稳定的、正的非常量平衡状态，或非平凡的稳定时间周期状态。出租车可以是令人厌恶的或有吸引力的，并且人口互动动态相当普遍。两种类型的模式形成都是作为单参数分叉解决方案从不稳定的恒定静止状态分支出来的。在没有我们的出租车机制的情况下，共存正稳态在合适的条件下是局部渐近稳定的。在存在足够强的排斥猎物防御的情况下，将形成模式。然而，在有吸引力的出租车案例中，吸引力需要足够弱才能形成图案。我们的方法是 Crandall-Rabinowitz 和 Hopf 分岔理论的应用。我们确定这两种类型的分支的存在，并开发表达式来确定它们的稳定性。