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Turing instability and Hopf bifurcation induced by prey refuge in a diffusive predator–prey system with stage structure and anti-predation

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Abstract

This paper studies a class of reaction-diffusion predator–prey system with stage-structured and anti-predation behavior. Based on the original deterministic system, we consider the prey refuge and study how it affects the stability of the populations. We first give the boundedness of solutions and then investigate the stability of equilibrium points with biological significance in the temporal system and spatial system. Further, the prey refuge is regarded as a bifurcation parameter for discussing the relevant Hopf bifurcation properties. Moreover, theoretical results show that the diffusion can cause Turing instability, interestingly, we find that a large diffusion rate of the prey can cause Turing instability, while large diffusion rates of predators would inhibit the appearance of Turing instability. Biologically, moderate anti-predation behavior among populations can promote system stability.

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References

  1. D.X. Geng, W.H. Jiang, Y. Lou, H.B. Wang, Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition. Stud. Appl. Math. 148(1), 396–432 (2022)

    Article  MathSciNet  Google Scholar 

  2. S.Q. Zhang, S.L. Yuan, T.H. Zhang, A predator–prey model with different response functions to juvenile and adult prey in deterministic and stochastic environments. Appl. Math. Comput. 413(15), 126598 (2022)

    MathSciNet  MATH  Google Scholar 

  3. C.H. Wang, S.L. Yuan, H. Wang, Spatiotemporal patterns of a diffusive prey–predator model with spatial memory and pregnancy period in an intimidatory environment. J. Math. Biol. 84(3), 12 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Kumar, R. Kumar, C. Cattani, B. Samet, Chaotic behaviour of fractional predator–prey dynamical system. Chaos Solitons Fract. 135, 109811 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. A.J. Lotka, Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. 6, 410–415 (1920)

    Article  ADS  Google Scholar 

  6. V. Volterra, Variazioni e fluttuazioni del numero individui in specie animali conviventi. Mem. Acad. Lincei Roma 2, 31–113 (1926)

    MATH  Google Scholar 

  7. Y.T. Cai, C.C. Wang, D.J. Fan, Bifurcation analysis of a predator–prey model with age structure. Int. J. Bifurc. Chaos 30(08), 2050114 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. M.M. Chen, R. Yuan, Optimal harvesting for a predator-prey system with age structure and reserve area. Math. Methods Appl. Sci. 45, 11849–11874 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  9. P. Yang, Hopf bifurcation of an age-structured prey–predator model with Holling type II functional response incorporating a prey refuge. Nonlinear Anal. Real. 49, 368–385 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Dubey, A. Kumar, Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays. Nonlinear Dyn. 96, 2653–2679 (2019)

    Article  MATH  Google Scholar 

  11. Y. Wang, X.F. Zou, On a predator–prey system with digestion delay and anti-predation strategy. J. Nonlinear Sci. 30, 1579–1605 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. K.D. Prasad, B.S.R.V. Prasad, Qualitative analysis of additional food provided predator–prey system with anti-predator behaviour in prey. Nonlinear Dyn. 96, 1765–1793 (2019)

    Article  MATH  Google Scholar 

  13. R.Z. Yang, J. Ma, Analysis of a diffusive predator-prey system with anti-predator behaviour and maturation delay. Chaos Solitons Fract. 109, 128–139 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. F. Faraji, A. Janssen, M.W. Sabelis, Oviposition patterns in a predatory mite reduce the risk of egg predation caused by prey. Ecol. Entomol. 07, 660–664 (2002)

    Article  Google Scholar 

  15. K. Able, S. Hagan, K. Kovitvongsa, S. Brown, J. Lamonaca, Piscivory by the mummichog (fundulus heteroclitus): evidence from the laboratory and saltmarshes. J. Exp. Mar. Biol. Ecol. 345, 26–37 (2007)

    Article  Google Scholar 

  16. R. Kneib, Testing for indirect effects of predation in an intertidal soft-bottom community. Ecology 69, 1795–1805 (1998)

    Article  Google Scholar 

  17. J. Li, X.H. Liu, C.J. Wei, The impact of role reversal on the dynamics of predator-prey model with stage structure. Appl. Math. Model 104, 339–357 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Aoki, U. Kurosu, S. Usuba, First instar larvae of the sugar-cane wooly aphid, Ceratovacuna lanigera (Homotera, Pemphigidae), attack its predators. Kontyu 52, 458–460 (1984)

    Google Scholar 

  19. W.D. Wang, L.S. Wang, A predator–prey system with stage-structure for predator. Comput. Math. Appl. 33, 83–91 (1997)

    MathSciNet  Google Scholar 

  20. T.T. Ma, X.Z. Meng, T. Hayat, A. Hobiny, Hopf bifurcation induced by time delay and inflfluence of Allee effect in a diffusive predator–prey system with herd behavior and prey chemotaxis. Nonlinear Dyn. 108, 4581–4598 (2022)

    Article  Google Scholar 

  21. J. Wang, Y.L. Cai, S.M. Fu, W.M. Wang, The effect of the fear factor on the dynamics of a predator–prey model incorporating the prey refuge. Chaos 29, 083109 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. M. Das, G.P. Samanta, A delayed fractional order food chain model with fear effect and prey refuge. Math. Comput. Simul. 178, 237–267 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Y.L. Song, H.P. Jiang, Y. Yuan, Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator–prey model. J. Appl. Anal. Comput. 9, 1132–1164 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Y.L. Song, H.P. Jiang, Q.X. Liu, Y. Yuan, Spatiotemporal dynamics of the diffusive mussel-algae model near Turing-Hopf bifurcation. SIAM J. Appl. Dyn. Syst. 16(4), 2030–2062 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Mishra, S.N. Raw, B. Tiwari, On a cannibalistic predator–prey model with prey defense and diffusion. Appl. Math. Model. 90, 165–190 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Chakraborty, N. Bairagi, Complexity in a prey–predator model with prey refuge and diffusion. Ecol. Complex. 37, 11–23 (2019)

    Article  Google Scholar 

  27. J. Murray, Mathematical biology: I. An introduction (Springer, 2002)

    Book  MATH  Google Scholar 

  28. T.K. Kar, Stability analysis of a prey–predator model incorporating a prey refuge. Commun. Nonlinear Sci. 10, 681–691 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. J.P. Shi, C.C. Wang, H. Wang, Spatial movement with diffusion and memory-based self-diffusion and cross-diffusion. J. Differ. Equ. 305, 242–269 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Y.L. Song, S.H. Wu, H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect. J. Differ. Equ. 267, 6316–6351 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Y.L. Song, Y. Peng, T.H. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system. J. Differ. Equ. 300, 597–624 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12271308), the Shandong Provincial Natural Science Foundation (No. ZR2019MA003), the Research Fund for the Taishan Scholar Project of Shandong Province of China.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ludong University, Yantai, 264025, People’s Republic of China

    Tingting Ma

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Xinzhu Meng

  3. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, 21589, Jeddah, Saudi Arabia

    Xinzhu Meng & Abdullah Khames Alzahrani

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  1. Tingting Ma
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Correspondence to Xinzhu Meng.

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Ma, T., Meng, X. & Alzahrani, A.K. Turing instability and Hopf bifurcation induced by prey refuge in a diffusive predator–prey system with stage structure and anti-predation. Eur. Phys. J. Plus 138, 624 (2023). https://doi.org/10.1140/epjp/s13360-023-04243-3

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