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Dynamic Analysis of a Diffusive Predator–Prey Model with Hunting Cooperation Functional Response and Prey-Taxis

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Abstract

Prey-taxis shows the tendency of predator moving toward the direction of gradient of prey density function. It is well known that it plays an important role in the study of biological populations. In this paper, we introduce prey-taxis into a diffusive predator–prey model with hunting cooperation functional response. First, we investigate the effects of prey-taxis on the stability of the positive equilibrium. The results show that there exists Turing instability when the prey-taxis is less than the critical value, and the positive equilibrium is locally asymptotically stable when prey-taxis is larger than the critical value. Then, we prove the existence of nonconstant positive steady states bifurcating from the positive equilibrium by using the bifurcation theory. Finally, our theoretical analyses are illustrated by numerical simulations.

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References

  1. Zhang, S., Yuan, S., Zhang, T.: Dynamic analysis of a stochastic eco-epidemiological model with disease in predators. Stud. Appl. Math. 149(1), 5–42 (2022)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  4. Tripathi, J.P., Bugalia, S., Jana, D., Gupta, N., Tiwari, V., Li, J., Sun, G.-Q.: Modeling the cost of anti-predator strategy in a predator–prey system: the roles of indirect effect. Math. Methods Appl. Sci. 45(8), 4365–4396 (2022)

    MathSciNet  ADS  Google Scholar 

  5. Tripathi, J.P., Jana, D., Vyshnavi Devi, N.S.N.V.K., Tiwari, V., Abbas, S.: Intraspecific competition of predator for prey with variable rates in protected areas. Nonlinear Dyn. 102(1), 511–535 (2020)

    Google Scholar 

  6. Tripathi, J.P., Abbas, S., Sun, G.-Q., Jana, D., Wang, C.-H.: Interaction between prey and mutually interfering predator in prey reserve habitat: pattern formation and the Turing–Hopf bifurcation. J. Franklin Inst. Eng. Appl. Math. 355(15), 7466–7489 (2018)

    MathSciNet  Google Scholar 

  7. Tripathi, J.P., Abbas, S., Thakur, M.: Dynamical analysis of a prey-predator model with Beddington–DeAngelis type function response incorporating a prey refuge. Nonlinear Dyn. 80(1–2), 177–196 (2015)

    MathSciNet  Google Scholar 

  8. Tripathi, J.P., Abbas, S., Thakur, M.: A density dependent delayed predator-prey model with Beddington–DeAngelis type function response incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 427–450 (2015)

    MathSciNet  ADS  Google Scholar 

  9. Schmidt, P., Mech, L.: Wolf pack size and food acquisition. Am. Nat. 150(4), 513–517 (1997)

    CAS  PubMed  Google Scholar 

  10. Creel, S., Creel, N.: Communal hunting and pack size in African wild dogs, lycaon-pictus. Anim. Behav. 50(5), 1325–1339 (1995)

    Google Scholar 

  11. Hector, D.: Cooperative hunting and its relationship to foraging succesa and prey size in an avian predator. Ethology 73(3), 247–257 (1986)

    Google Scholar 

  12. Bshary, R., Hohner, A., Ait-el Djoudi, K., Fricke, H.: Interspecific communicative and coordinated hunting between groupers and giant moray eels in the red sea. PLoS Biol. 4(12), 2393–2398 (2006)

    CAS  Google Scholar 

  13. Cosner, C., DeAngelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on the functional response of predators. Theor. Popul. Biol. 56(1), 65–75 (1999)

    CAS  PubMed  Google Scholar 

  14. Ryu, K., Ko, W., Haque, M.: Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities. Nonlinear Dyn. 94, 1639–1656 (2018)

    Google Scholar 

  15. Song, Y., Zhang, T., Peng, Y.: Turing–Hopf bifurcation in the reaction–diffusion equations and its applications. Commun. Nonlinear Sci. Numer. Simul. 33, 229–258 (2016)

    MathSciNet  ADS  Google Scholar 

  16. Song, Y., Jiang, H., Yuan, Y.: Turing–Hopf bifurcation in the reaction–diffusion system with delay and application to a diffusive predator-prey model. J. Appl. Anal. Comput. 9(3), 1132–1164 (2019)

    MathSciNet  Google Scholar 

  17. Chen, S., Wei, J., Yu, J.: Stationary patterns of a diffusive predator–prey model with Crowley–Martin functional response. Nonlinear Anal. Real World Appl. 39, 33–57 (2018)

    MathSciNet  CAS  Google Scholar 

  18. Ni, W., Wang, M.: Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey. J. Differ. Equ. 261(7), 4244–4274 (2016)

    MathSciNet  ADS  Google Scholar 

  19. Wang, J., Wei, J., Shi, J.: Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems. J. Differ. Equ. 260(4), 3495–3523 (2016)

    MathSciNet  ADS  Google Scholar 

  20. Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J. Differ. Equ. 246(5), 1944–1977 (2009)

    MathSciNet  ADS  Google Scholar 

  21. Huang, J., Lu, G., Ruan, S.: Existence of traveling wave solutions in a diffusive predator-prey model. J. Math. Biol. 46(2), 132–152 (2003)

    MathSciNet  PubMed  Google Scholar 

  22. Li, W.-T., Wu, S.-L.: Traveling waves in a diffusive predator-prey model with Holling type-III functional response. Chaos Solitons Fractals 37(2), 476–486 (2008)

    MathSciNet  ADS  Google Scholar 

  23. Zhang, T., Liu, X., Meng, X., Zhang, T.: Spatio-temporal dynamics near the steady state of a planktonic system. Comput. Math. Appl. 75(12), 4490–4504 (2018)

    MathSciNet  Google Scholar 

  24. Kuto, K.: Stability of steady-state solutions to a prey-predator system with cross-diffusion. J. Differ. Equ. 197(2), 293–314 (2004)

    MathSciNet  ADS  Google Scholar 

  25. Kuto, K., Yamada, Y.: Multiple coexistence states for a prey-predator system with cross-diffusion. J. Differ. Equ. 197(2), 315–348 (2004)

    MathSciNet  ADS  Google Scholar 

  26. Peng, R., Wang, M., Yang, G.: Stationary patterns of the Holling–Tanner prey–predator model with diffusion and cross-diffusion. Appl. Math. Comput. 196(2), 570–577 (2008)

    MathSciNet  Google Scholar 

  27. Wang, Y.-X., Li, W.-T.: Spatial patterns of the Holling–Tanner predator–prey model with nonlinear diffusion effects. Appl. Anal. 92(10), 2168–2181 (2013)

    MathSciNet  Google Scholar 

  28. Zhou, J., Kim, C.-G., Shi, J.: Positive steady state solutions of a diffusive Leslie–Gower predator–prey model with Holling type II functional response and cross-diffusion. Discret. Contin. Dyn. Syst. 34(9), 3875–3899 (2014)

    MathSciNet  Google Scholar 

  29. Jia, D., Zhang, T., Yuan, S.: Pattern dynamics of a diffusive toxin producing phytoplankton–zooplankton model with three-dimensional patch. Int. J. Bifurc. Chaos 29, 1930011 (2019)

    MathSciNet  Google Scholar 

  30. Golovin, A.A., Matkowsky, B.J., Volpert, V.A.: Turing pattern formation in the brusselator model with superdiffusion. SIAM J. Appl. Math. 69(1), 251–272 (2008)

    MathSciNet  Google Scholar 

  31. Tzou, J., Matkowsky, B.J., Volpert, V.A.: Interaction of Turing and Hopf modes in the superdiffusive brusselator model. Appl. Math. Lett. 22(9), 1432–1437 (2009)

    MathSciNet  Google Scholar 

  32. Zhang, L., Tian, C.: Turing pattern dynamics in an activator-inhibitor system with superdiffusion. Phys. Rev. E 90(6), 062915 (2014)

    CAS  ADS  Google Scholar 

  33. Wu, S., Song, Y.: Stability analysis of a diffusive predator–prey model with hunting cooperation. J. Nonlinear Model. Anal. 3(2), 321–334 (2021)

    Google Scholar 

  34. Yang, R., Zhang, X., Jin, D.: Spatiotemporal dynamics in a delayed diffusive predator-prey system with nonlocal competition in prey and schooling behavior among predators. Bound. Value Probl. 2022(1), 56 (2022)

    MathSciNet  Google Scholar 

  35. Yan, S., Jia, D., Zhang, T., Yuan, S.: Pattern dynamics in a diffusive predator–prey model with hunting cooperations. Chaos Solitons Fractals 130, 109428 (2020)

    MathSciNet  Google Scholar 

  36. Djilali, S., Cattani, C.: Patterns of a superdiffusive consumer-resource model with hunting cooperation functional response. Chaos Solitons Fractals 151, 111258 (2021)

    MathSciNet  Google Scholar 

  37. Kareiva, P., Odell, G.: Swarms of predators exhibit" preytaxis" if individual predators use area-restricted search. Am. Nat. 130(2), 233–270 (1987)

    Google Scholar 

  38. Grünbaum, D.: Using spatially explicit models to characterize foraging performance in heterogeneous landscapes. Am. Nat. 151(2), 97–113 (1998)

    PubMed  Google Scholar 

  39. Chakraborty, A., Singh, M., Lucy, D., Ridland, P.: Predator–prey model with prey-taxis and diffusion. Math. Comput. Model. 46(3–4), 482–498 (2007)

    MathSciNet  Google Scholar 

  40. Sapoukhina, N., Tyutyunov, Y., Arditi, R.: The role of prey taxis in biological control: a spatial theoretical model. Am. Nat. 162, 61–76 (2003)

    PubMed  Google Scholar 

  41. Ainseba, B., Bendahmane, M., Noussair, A.: A reaction–diffusion system modeling predator–prey with prey-taxis. Nonlinear Anal. Real World Appl. 9(5), 2086–2105 (2008)

    MathSciNet  Google Scholar 

  42. Lee, J.M., Hillen, T., Lewis, M.A.: Pattern formation in prey-taxis systems. J. Biol. Dyn. 3(6), 551–573 (2009)

    MathSciNet  CAS  PubMed  Google Scholar 

  43. Tao, Y.: Global existence of classical solutions to a predator–prey model with nonlinear prey-taxis. Nonlinear Anal. Real World Appl. 11(3), 2056–2064 (2010)

    MathSciNet  Google Scholar 

  44. Wang, Q., Song, Y., Shao, L.: Nonconstant positive steady states and pattern formation of 1d prey-taxis systems. J. Nonlinear Sci. 27(1), 71–97 (2017)

    MathSciNet  ADS  Google Scholar 

  45. Wang, X., Wang, W., Zhang, G.: Global bifurcation of solutions for a predator–prey model with prey-taxis. Math. Methods Appl. Sci. 38(3), 431–443 (2015)

    MathSciNet  ADS  Google Scholar 

  46. Song, Y., Tang, X.: Stability, steady state bifurcations, and turing patterns in a predator–prey model with herd behavior and prey-taxis. Stud. Appl. Math. 139(3), 371–404 (2017)

    MathSciNet  Google Scholar 

  47. Li, C., Wang, X., Shao, Y.: Steady states of a predator–prey model with prey-taxis. Nonlinear Anal. Theory Methods Appl. 97, 155–168 (2014)

    MathSciNet  Google Scholar 

  48. Lee, J.M., Hillen, T., Lewis, M.A.: Continuous traveling waves for prey-taxis. Bull. Math. Biol. 70(3), 654–676 (2008)

    MathSciNet  CAS  PubMed  Google Scholar 

  49. Wang, J., Wu, S., Shi, J.: Pattern formation in diffusive predator–prey systems with predator-taxis and prey-taxis. Discrete Contin. Dyn. Syst.-Ser. B 26(3), 1273–1289 (2021)

    MathSciNet  Google Scholar 

  50. Shi, J., Wang, X.: On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246(7), 2788–2812 (2009)

    MathSciNet  ADS  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and valuable suggestions which greatly improved the initial manuscript. This research was supported by Natural Science Foundation of Shanghai (No. 23ZR1401700) and National Natural Science Foundation of China (Nos. 12271088 and 12271308).

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Authors and Affiliations

  1. Department of Mathematics, Donghua University, Shanghai, 201620, People’s Republic of China

    Yahong Peng & Xingyu Yang

  2. Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia

    Tonghua Zhang

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  1. Yahong Peng
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All authors wrote and revised the main manuscript text and Y.P. and X.Y. prepared figures 1–3 and the calculations. All authors reviewed the manuscript.

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Correspondence to Tonghua Zhang.

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Peng, Y., Yang, X. & Zhang, T. Dynamic Analysis of a Diffusive Predator–Prey Model with Hunting Cooperation Functional Response and Prey-Taxis. Qual. Theory Dyn. Syst. 23, 64 (2024). https://doi.org/10.1007/s12346-023-00914-9

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