关键词: 恐惧, 狩猎合作, 跨临界分岔, Hopf分岔, Turing斑图

Abstract: In ecosystems, predator-induced fear can suppress prey reproduction, while cooperative behavior among species is a widespread phenomenon. This paper establishes a predator-prey model incorporating fear effects, hunting cooperation, and harvesting. For the non-spatial model, we prove the positivity and boundedness of solutions, derive sufficient conditions for the existence and local stability of all equilibrium points, and demonstrate the global stability of the interior equilibrium by constructing an appropriate Lyapunov function. Furthermore, we explore local bifurcations with respect to key parameters: trans-critical bifurcation is proven via Sotomayor's theorem, and Hopf bifurcation is analyzed under the hunting cooperation parameter. For the spatial model, we provide a detailed stability analysis, investigate the conditions for Turing instability, and identify various Turing patterns. The biological implications of these patterns in the two-dimensional spatial model are discussed. Finally, numerical simulations are conducted to validate the analytical findings for both non-spatial and spatial models.

Key words: fear, hunting cooperation, transcritical bifurcation, Hopf bifurcation, Turing pattern