在生物数学中, 捕食-食饵系统始终是研究的热点. 目前, 捕食者觅食过程中的干扰行为对捕食-食饵系统动力学的影响已经受到了广泛的关注, 然而, 捕食者觅食过程中的合作行为在很大程度上被忽略了. 本文主要考虑具有合作捕食效应的捕食-食饵系统的共存问题, 讨论合作捕食对捕食-食饵系统的影响.

       本文第一部分研究了一类齐次 Neumann 边界条件下具有空间异质和合作捕食的捕食-食饵模型. 首先, 利用线性化理论的谱分析和比较原理, 证明了平凡解和半平凡解的全局渐近稳定性; 其次, 利用不动点指数理论, 建立了共存解存在的充分条件; 最后, 利用 MATLAB 进行数值模拟, 验证已得到的理论结果, 并进一步探讨空间异质和合作捕食对共存解的影响. 结果表明, 空间异质和合作捕食均会对两物种的共存产生明显影响.

       本文第二部分研究了一类齐次 Dirichlet 边界条件下具有交叉扩散和合作捕食的捕食-食饵模型. 首先, 基于线性化理论的谱分析, 证明了平凡解和半平凡解具有局部渐近稳定性; 进一步, 利用锥上的度理论, 给出了共存解存在的充分条件; 最后, 利用 MATLAB 进行数值模拟, 讨论交叉扩散和合作捕食对共存解的影响. 结果表明, 交叉扩散和合作捕食均会对两物种的共存产生明显影响.

       In biomathematics, predator-prey systems have always been a hot spot of research. At present, the effect of predator interference behavior during foraging on the dynamics of predator-prey systems has received extensive attention. However, the cooperative behavior of predators during foraging has been largely ignored. In this paper, the coexistence problem of predator-prey systems with cooperative predation effect is considered, and the effect of hunting cooperation on predator-prey systems is discussed.

       In the first part of this paper, a predator-prey model with spatial heterogeneity and hunting cooperation under homogeneous Neumann boundary conditions is investigated. Firstly, by using the spectral analysis of linearized theory and the comparison principle, the global asymptotic stability of trivial solution and semi-trivial solutions is obtained. Secondly, the sufficient conditions for the existence of coexistence solutions are established by the fixed point index theory. Finally, numerical simulations are carried out with MATLAB to verify the theoretical results obtained and further discass the effect of spatial heterogeneity and hunting cooperation on coexistence solutions. The results indicate that both spatial heterogeneity and hunting cooperation have obvious influence on the coexistence of the two species.

       In the second part of this paper, a predator-predator model with cross-diffusion and hunting cooperation under homogeneous Dirichlet boundary conditions is studied. Based on the spectral analysis of linearized theory, the local asymptotic stability of trivial and semi-trivial solutions is proved. Moreover, the sufficient conditions for the existence of coexistence solutions are given by using the degree theory in cones. Finally, numerical simulations are performed with MATLAB to discuss the effect of cross-diffusion and hunting cooperation on coexistence solutions. The results suggest that cross-diffusion and hunting cooperation have obvious effect on the coexistence of the two species.

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