In this manuscript, we investigate the fear effect on the spatiotemporal behavior of the predator–prey model with prey social behavior and cross-diffusive. Our main interest is to determine the existence of Turing patterns and the fear effect performed by predators on the prey population and the group defense. The fear can lead to a partition of the prey herd which is known as prey escaping. It is obtained that the system has rich dynamics elaborated by the presence of Turing patterns and Turing–Hopf bifurcation. The nature of Turing patterns is successfully discussed by analyzing the amplitude equations with a multiple-timescale technique. After studying the stability of these amplitude equations, it has been identified various Turing patterns driven by the cross-diffusion. Further, the effect of the fear rate or escaping rate on the behavior of the solution is discussed. The theoretical results are checked numerically.

在本手稿中，我们研究了具有食饵社交行为和交叉扩散的食饵-食饵模型时空行为的恐惧效应。我们的主要兴趣是确定图灵模式的存在以及捕食者对猎物种群和群体防御的恐惧效应。恐惧会导致猎物群的分裂，这被称为猎物逃逸。结果表明，该系统具有丰富的动力学特性，这是由于存在Turing模式和Turing-Hopf分叉所引起的。通过使用多时标技术分析振幅方程，成功地讨论了图灵模式的性质。在研究了这些振幅方程的稳定性之后，已经确定了由交叉扩散驱动的各种图灵模式。更多，讨论了恐惧率或逃避率对解决方案行为的影响。对理论结果进行数值检验。