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Hopf Bifurcation Analysis of a Predator–Prey Model with Prey Refuge and Fear Effect Under Non-diffusion and Diffusion

  • Haisu Zhang [2] ; Haokun Qi [1]
    1. [1] Anshan Normal University

      Anshan Normal University

      China

    2. [2] Taishan College of Science and Technology (China)
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 4, 2023
  • Idioma: inglés
  • Enlaces
  • Resumen

    • In this paper, we propose a predator–prey model with prey refuge and fear effect under non-diffusion and diffusion. For the non-diffusion ODE model, we first analyze the existence and stability of equilibria. Then, the existence of transcritical bifurcation, Hopf bifurcation and limit cycle is discussed, respectively. We find that when the cost of minimum fear \eta is taken as the bifurcation parameter, it not only influence the occurrence of Hopf bifurcation but also alters its direction. For diffusion predator–prey model under homogeneous Neumann boundary conditions, we observe that the Turing instability does not occur, but the Hopf bifurcation will manifest near the interior equilibrium. By considering \eta as the bifurcation parameter, the direction and stability of spatially homogeneous periodic orbits are established. At last, the validity of the theoretical analysis are verified by a series of numerical simulations. The results indicate that prey refuge and fear effect play an key role in the stability of populations.

  • Referencias bibliográficas
    • 1. Ritchie, E.G., Johnson, C.N.: Predator interactions, mesopredator release and biodiversity conservation. Ecol. Lett. 12(9), 982–998 (2009)
    • 2. Holling, C.S.: The functional response of invertebrate predators to prey density. Mem. Entomol. Soc. Can. 98, 5–86 (1966)
    • 3. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340...
    • 4. DeAngelis, D.L., Goldstein, R.A., Oneill, R.V.: A model for tropic interaction. Ecology 56, 881–892 (1975)
    • 5. Abrams, P.A., Ginzburg, L.R.: The nature of predation: prey dependent, ratio dependent or neither? Trends Ecol. Evol. 15(8), 337–341 (2000)
    • 6. Ivlev, V.S.: Experimental ecology of the feeding of fishes, University Microfilms, (1961)
    • 7. Hassell, M.P., Varley, G.C.: New inductive population model for insect parasites and its bearing on biological control. Nature 223, 1133–1137...
    • 8. Hu, D., Li, Y., Liu, M., Bai, Y.: Stability and Hopf bifurcation for a delayed predator-prey model with stage structure for prey and Ivlev-type...
    • 9. Hu, D., Cao, H.: Stability and bifurcation analysis in a predator-prey system with Michaelis–Menten type predator harvesting. Nonlinear...
    • 10. Hu, D., Cao, H.: Bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type. Commun. Nonlinear Sci. 22(1–3),...
    • 11. Hu, D., Yu, X., Zheng, Z., Zhang, C., Liu, M.: Multiple bifurcations in a discrete bazykin predator-prey model with predator intraspecific...
    • 12. Zhang, W., Zhao, S., Meng, X., Zhang, T.: Evolutionary analysis of adaptive dynamics model under variation of noise environment. Appl....
    • 13. Xiang, C., Huang, J., Ruan, S., Xiao, D.: Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response....
    • 14. Qi, H., Meng, X.: Threshold behavior of a stochastic predatorCprey system with prey refuge and fear effect. Appl. Math. Lett. 113, 106846...
    • 15. Qi, H., Meng, X., Hayat, T., Hobiny, A.: Stationary distribution of a stochastic predator-prey model with hunting cooperation. Appl. Math....
    • 16. Liu, Q., Jiang, D.: Influence of the fear factor on the dynamics of a stochastic predator-prey model. Appl. Math. Lett. 112, 106756 (2021)
    • 17. Zhang, H., Cai, Y., Fu, S., Wang, W.: Impact of the fear effect in a prey-predator model incorporating a prey refuge. Appl. Math. Comput....
    • 18. McNair, J.N.: Stability effects of prey refuges with entry-exit dynamics. J. Theor. Biol. 125, 449–464 (1987)
    • 19. Kar, T.K.: Stability analysis of a prey-predator model incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simulat. 10, 681–691...
    • 20. Chen, L., Chen, F., Chen, L.: Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant...
    • 21. Ma, Z., Wang, S., Li, W., Li, Z.: The effect of prey refuge in a patchy predator-prey system. Math. Biosci. 243(1), 126–130 (2013)
    • 22. Zhou, Y., Sun, W., Song, Y., Zheng, Z., Lu, J., Chen, S.: Hopf bifurcation analysis of a predator-prey model with Holling-II type functional...
    • 23. Chang, X., Wei, J.: Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Math. Biosci. Eng....
    • 24. Altendorf, K.B., Laundré, J.W., González, C.A.L., Brown, J.S.: Assessing effects of predation risk on foraging behavior of mule deer....
    • 25. Laundré, J.W., Hernández, L., Altendorf, K.B.: Wolves, elk, and bison: reestablishing the landscape of fear in yellowstone National Park,...
    • 26. Creel, S., Christianson, D., Liley, S., Winnie, J.A.: Predation risk affects reproductive physiology and demography of elk. Science 315,...
    • 27. Zanette, L.Y., White, A.F., Allen, M.C., Clinchy, M.: Perceived predation risk reduces the number of offspring songbirds produce per year....
    • 28. Cresswell, W.: Predation in bird populations. J. Ornithol. 152(1), 251–263 (2011)
    • 29. Wang, X., Zanette, L., Zou, X.: Modelling the fear effect in predator-prey interactions. J. Math. Biol. 73, 1179–1204 (2016)
    • 30. Wang, X., Zou, X.: Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators. Bull. Math. Biol. 79(6),...
    • 31. Sarkara, K., Khajanchi, S.: Impact of fear effect on the growth of prey in a predator-prey interaction model. Ecol. Complex. 42, 100826...
    • 32. Sasmal, S.K., Takeuchi, Y.: Dynamics of a predator-prey system with fear and group defense. J. Math. Anal. Appl. 481(11), 123471 (2020)
    • 33. Roy, J., Alam, S.: Fear factor in a prey-predator system in deterministic and stochastic environment. Physica A. 541, 123359 (2020)
    • 34. Ko, W., Ryu, K.: Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. J....
    • 35. Peng, R., Shi, J.: Non-existence of non-constant positive steady states of two Holling type-II predatorprey systems: strong interaction...
    • 36. Li, C., Wang, X., Shao, Y.: Steady states of a predator-prey model with prey-taxis. Nonlinear Anal. 97, 155–168 (2014)
    • 37. Qiao, T., Cai, Y., Fu, S., Wang, W.: Stability and hopf bifurcation in a predator-prey model with the cost of anti-predator behaviors....
    • 38. Song, Y., Jiang, H., Yuan, Y.: Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey...
    • 39. Upadhyay, R.K., Roy, P., Datta, J.: Complex dynamics of ecological systems under nonlinear harvesting: Hopf bifurcation and turing instability....
    • 40. Tiwari, V., Tripathi, J.P., Mishra, S., Upadhyay, R.K.: Modeling the fear effect and stability of nonequilibrium patterns in mutually...
    • 41. Song, D., Li, C., Song, Y.: Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation...
    • 42. Li, Q., Liu, Z., Yuan, S.: Cross-diffusion induced Turing instability for a competition model with saturation effec. Appl. Math. Comput....
    • 43. Yan, S., Jia, D., Zhang, T., Yuan, S.: Pattern dynamics in a diffusive predator-prey model with hunting cooperations. Chaos Soliton. Fract....
    • 44. Song, Y., Tang, X.: Stability, steady-state bifurcations, and turing patterns in a predator-prey model with herd behavior and prey-taxis....
    • 45. Yang, R., Jin, D., Wang, W.: A diffusive predator-prey model with generalist predator and time delay. AIMS Math. 7(3), 4574–4591 (2022)
    • 46. Duan, D., Niu, B., Wei, J.: Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predatorprey model with fear effect, Chaos....
    • 47. Qi, H., Meng, X., Hayat, T., Hobiny, A.: Bifurcation dynamics of a reaction-diffusion predator-prey model with fear effect in a predator-poisoned...
    • 48. Lou, Y., Ni, W.: Diffusion, self-diffusion and cross-diffusion. J. Differ. Equations 131, 79–131 (1996)
    • 49. Yang, R., Nie, C., Jin, D.: Spatiotemporal dynamics induced by nonlocal competition in a diffusive predator-prey system with habitat complexity....
    • 50. Yang, R.,Wang, F., Jin, D.: Spatially inhomogeneous bifurcating periodic solutions induced by nonlocal competition in a predator-prey...
    • 51. Sotomayor, J.: Generic bifurcation of dynamical system. Dynam Syst (1973). https://doi.org/10.1016/ B978-0-12-550350-1.50047-3
    • 52. Meiss JD.: Differential Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia (2007)
    • 53. Wiggins, S., Golubitsky, M.: Introduction to applied nonlinear dynamical systems and chaos, vol. 2. Springer, Berlin (1990)
    • 54. Hassard, B.D., Kazarinoff, N.D., Wan, Y.: Theory and applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)
    • 55. Garvie, M.R.: Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB. Bull. Math. Biol....
    • 56. Burton, T.A.: Volterra integral and differential equations. Academic Press, Inc., Orlando (1983)
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