When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. We formulate a patch model to investigate the aforementioned fear effect which is indirect, in contrast to the predation as a direct effect, of the predator on the prey population. We consider not only cost but also benefit of anti-predation response of the prey, and explore their trade-offs together as well as the impact of the fear effect mediated dispersals of the prey. In the case of constant response level, if there is no dispersal and for some given response functions, the model indicates the existence of an evolutionary stable strategy which is also a convergence stable strategy for the response level; and if there is dispersal, the analysis of the model shows that it will enhance the co-persistence of the prey on both patches. Considering the trait as another variable, we continue to study the evolution of anti-predation strategy for the model with dispersal, which leads to a three-dimensional system of ordinary differential equations. We perform some numerical simulations, which demonstrate global convergence to a positive equilibrium with the response level evolving towards a positive constant level, implying the existence of an optimal anti-predation response level.

当感知到来自捕食者的风险时，猎物可能会通过减少繁殖和减少或增加（取决于物种）其活动性来做出反应。我们制定了一个补丁模型来研究上述的恐惧效应，与捕食者对猎物种群的直接影响相反，这种恐惧效应是间接的。我们不仅考虑了猎物反捕食反应的成本而且还考虑了收益，并一起探讨了它们的权衡以及恐惧效应介导的猎物扩散的影响。在响应水平不变的情况下，如果不存在离散，并且对于某些给定的响应函数，模型表明存在进化稳定策略，这也是响应水平的收敛稳定策略；如果有分散，该模型的分析表明，它将增强猎物在两个补丁上的协同持久性。考虑到性状作为另一个变量，我们继续研究具有离散性的模型的反捕食策略的演化，这导致了常微分方程的三维系统。我们进行了一些数值模拟，这表明全局收敛到正平衡，响应水平向正恒定水平发展，这意味着存在最佳的反捕食响应水平。