In this paper, we consider a new approach of prey escaping from herd in a predator-prey model with the presence of spatial diffusion. First, the sensitivity of the equilibrium state density with respect to the escaping rate has been studied. Then, the analysis of the non diffusive system was investigated where boundedness, local, global stability, Hopf bifurcation are obtained. Besides, for the diffusive system, we proved the occurrence of Hopf bifurcation and the non existence of diffusion driven instability. Furthermore, the direction of Hopf bifurcation has been proved using the normal form on the center manifold. Some numerical simulations have been used to illustrate the obtained results.
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.
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