This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value λ 1 D (Ω 0 ), and demonstrate that the existence of the predator in $${\overline \Omega _0}$$ Ω ¯ 0 only depends on the relationship of the growth rate μ of the predator and λ 1 D (Ω 0 ), not on the prey. Furthermore, when μ < λ 1 D (Ω 0 ), we obtain the existence and uniqueness of its positive steady state solution, while when μ ≥ λ 1 D (Ω 0 ), the predator and the prey cannot coexist in $${\overline \Omega _0}$$ Ω ¯ 0 . Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $${\overline \Omega _0}$$ Ω ¯ 0 , which is different from that of the classical Lotka-Volterra predator-prey model.

中文翻译：

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具有拥挤项的 Lotka-Volterra Predator-Prey 模型正稳态解的存在唯一性

本文讨论了在捕食者方程中具有拥挤项的 Lotka-Volterra 捕食者-猎物模型。我们得到了一个临界值 λ 1 D (Ω 0 )，并证明了捕食者在 $${\overline \Omega _0}$$ Ω¯ 0 中的存在仅取决于捕食者的增长率 μ 和λ 1 D (Ω 0 )，不在猎物上。此外，当 μ < λ 1 D (Ω 0 ) 时，我们得到其正稳态解的存在唯一性，而当 μ ≥ λ 1 D (Ω 0 ) 时，捕食者和猎物不能在 $${\上划线 \Omega _0}$$ Ω ¯ 0 。我们的结果表明，猎物和捕食者的共存对拥挤区域 $${\overline \Omega _0}$$ Ω¯ 0 的大小很敏感，这与经典的 Lotka-Volterra 捕食者-猎物不同模型。