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effect of shape of herd behavior in a diffusive predator-prey model with time delay in predator
Keywords:predator-prey model, herd behavior, spatial diffusion, time delay, quadratic mortality, Turing instability, Turing-Hopf bifurcation.
      In this paper, we deal with the effect of the shape of herd behavior on the interaction between predator and prey. The model analysis was studied in three parts. The first, The analysis of the system in the absence of spatial diffusion and the time delay, where the local stability of the equilibrium states, the existence of Hopf bifurcation have been investigated. For the second part, the spatiotemporal dynamics introduce by self diffusion was determined, where the existence of Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been proved. Further, the order of Hopf bifurcation points and regions of the stability of the non trivial equilibrium was given. In the last part of the paper, we studied the delay effect on thestability of the non trivial equilibrium, where we proved that the delay can lead to the instability of interiour equilibrium state, and also the existence of Hopf bifurcation. A numerical simulation was carried out to insure the theoretical results.