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Existence and Uniqueness of the Positive Steady State Solution for a Lotka-Volterra Predator-Prey Model with a Crowding Term

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Abstract

This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value λ D1 0), and demonstrate that the existence of the predator in \({\overline \Omega _0}\) only depends on the relationship of the growth rate μ of the predator and λ D1 0), not on the prey. Furthermore, when μ < λ D1 0), we obtain the existence and uniqueness of its positive steady state solution, while when μ ≥ λ D1 0), the predator and the prey cannot coexist in \({\overline \Omega _0}\). Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region \({\overline \Omega _0}\), which is different from that of the classical Lotka-Volterra predator-prey model.

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Correspondence to Xianzhong Zeng.

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The work was supported by the Hunan Provincial Natural Science Foundation of China (2019JJ40079, 2019JJ50160), the Scientific Research Fund of Hunan Provincial Education Department (16A071, 19A179) and the National Natural Science Foundation of China (11701169)

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Zeng, X., Liu, L. & Xie, W. Existence and Uniqueness of the Positive Steady State Solution for a Lotka-Volterra Predator-Prey Model with a Crowding Term. Acta Math Sci 40, 1961–1980 (2020). https://doi.org/10.1007/s10473-020-0622-7

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  • DOI: https://doi.org/10.1007/s10473-020-0622-7

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