This work deals with a general cross-diffusion system modeling the dynamics behavior of two predators and one prey with signal-dependent diffusion and sensitivity subject to homogeneous Neumann boundary conditions. Firstly, in light of some \(L^p-\)estimate techniques, we rigorously prove the global existence and uniform boundedness of positive classical solutions in any dimensions with suitable conditions on motility functions and the coefficients of logistic source. Moreover, by constructing some appropriate Lyapunov functionals, we further establish the asymptotic behavior of solutions to a specific model with Lotka-Volterra type functional responses and density-dependent death rates for two predators as well as logistic type for the prey. Our results not only generalize the previously known one, but also present some new conclusions.

这项工作涉及一个通用的交叉扩散系统，该系统对两个捕食者和一个猎物的动力学行为进行建模，该系统具有信号相关的扩散和受均匀 Neumann 边界条件影响的敏感性。首先，根据一些\(L^p-\)估计技术，我们严格证明了在运动函数和逻辑源系数的合适条件下，任何维度的正经典解的全局存在性和统一有界性。此外，通过构建一些适当的 Lyapunov 泛函，我们进一步建立了具有 Lotka-Volterra 类型功能响应和两个捕食者的密度相关死亡率以及猎物的逻辑类型的特定模型的解的渐近行为。我们的结果不仅概括了先前已知的结果，而且还提出了一些新的结论。