In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

中文翻译：

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具有密度相关运动的捕食者-猎物系统的全局动力学和时空模式

在本文中，我们在具有 Neumann 边界条件的二维有界域中研究了具有密度依赖性猎物-趋向性的捕食者-猎物系统的全局有界性、渐近稳定性和模式形成，其中运动系数 (diffusiq'dfdon) 和捕食者的移动性（猎物-出租车）通过猎物密度依赖性运动功能相关联。我们通过构造 Lyapunov 泛函，在一定的参数条件下，建立了具有一致时间界限的经典解的存在以及空间均匀的仅猎物稳态和共存稳态的全局稳定性。通过数值模拟，我们进一步证明了空间均匀的时间周期模式，稳定的空间不均匀模式和混沌的时空模式对于稳定机制之外的参数都是可能的。我们还从数值模拟中发现，线性系统和非线性系统之间的时间动力学差异很大，猎物密度相关的运动函数可以触发模式形成。