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Stability and Turing Patterns of a Predator-prey Model with Holling Type II Functional Response and Allee Effect in Predator

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Abstract

In this paper, we are concerned with a predator-prey model with Holling type II functional response and Allee effect in predator. We first mathematically explore how the Allee effect affects the existence and stability of the positive equilibrium for the system without diffusion. The explicit dependent condition of the existence of the positive equilibrium on the strength of Allee effect is determined. It has been shown that there exist two positive equilibria for some modulate strength of Allee effect. The influence of the strength of the Allee effect on the stability of the coexistence equilibrium corresponding to high predator biomass is completely investigated and the analytically critical values of Hopf bifurcations are theoretically determined. We have shown that there exists stability switches induced by Allee effect. Finally, the diffusion-driven Turing instability, which can not occur for the original system without Allee effect in predator, is explored, and it has been shown that there exists diffusion-driven Turing instability for the case when predator spread slower than prey because of the existence of Allee effect in predator.

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Authors and Affiliations

  1. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

    Lu Chen, Feng Yang & Yong-li Song

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  1. Lu Chen
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  2. Feng Yang
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  3. Yong-li Song
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Correspondence to Yong-li Song.

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Conflict of Interest

The authors declare no conflict of interest.

The project is supported by the National Natural Science Foundation of China (No. 11971143; 12071105) and Zhejiang Provincial Natural Science Foundation of China (No.LZ23A010001).

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Chen, L., Yang, F. & Song, Yl. Stability and Turing Patterns of a Predator-prey Model with Holling Type II Functional Response and Allee Effect in Predator. Acta Math. Appl. Sin. Engl. Ser. 39, 675–695 (2023). https://doi.org/10.1007/s10255-023-1084-1

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  • DOI: https://doi.org/10.1007/s10255-023-1084-1

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