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Qualitative analysis of a diffusive Crowley–Martin predator–prey model: the role of nonlinear predator harvesting

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Abstract

In this paper, we discuss a diffusive predator–prey system with mutually interfering predator and nonlinear harvesting in predator with Crowley–Martin functional response. The mathematical analysis of the system starts with the existence and uniqueness of solution of the system using \(C_0\) semigroup. The analysis reflects that the upper bound of rate of predator harvesting for the coexistence of the species can be guaranteed. In addition, we establish the existence and nonexistence of non-constant positive steady state. Explicit conditions on predator harvesting are obtained for local and global stability of interior equilibrium and also for the existence and nonexistence of non-constant steady-state solution. We also investigate conditions for Turing instabilities of the diffusive system analytically. Our results show that the effort of harvesting (g) provides a threshold value for existence of non-constant positive stationary solution. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the system exhibits interesting patterns. Some biological implications of obtained theoretical results have also been discussed.

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Acknowledgements

The project is funded by the National Natural Science Foundation of China under Grants 11671241 and 11331009, Outstanding Young Talents Support Plan of Shanxi province, and Selective Financial Support for Scientific and Technological Activities of Overseas Students in Shanxi Province. The research work of Jai Prakash Tripathi is supported by SERB, DST project [grant: ECR/2017/002786].

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Authors and Affiliations

  1. Department of Applied Sciences and Humanities, K.N.I.T., Sultanpur, Uttar Pradesh, 311001, India

    Vandana Tiwari

  2. Department of Mathematics, Central University of Rajasthan, NH-8, Bandarsindri, Kishangarh, Ajmer, Rajasthan, 305817, India

    Jai Prakash Tripathi

  3. School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, 175001, India

    Syed Abbas

  4. Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Jin-Shan Wang, Gui-Quan Sun & Zhen Jin

  5. Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Taiyuan, 030006, Shanxi, People’s Republic of China

    Jin-Shan Wang, Gui-Quan Sun & Zhen Jin

  6. Department of Mathematics, North University of China, Taiyuan, 030051, Shanxi, People’s Republic of China

    Gui-Quan Sun

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  1. Vandana Tiwari
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  2. Jai Prakash Tripathi
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Tiwari, V., Tripathi, J.P., Abbas, S. et al. Qualitative analysis of a diffusive Crowley–Martin predator–prey model: the role of nonlinear predator harvesting. Nonlinear Dyn 98, 1169–1189 (2019). https://doi.org/10.1007/s11071-019-05255-4

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