Multiple Bifurcations in a Discrete Bazykin Predator–Prey Model with Predator Intraspecific Interactions and Ratio-Dependent Functional Response

• Dongpo Hu ^[1] ; Xiao Yu ^[1] ; Zhaowen Zheng ^[2] ; Chuan Zhang ^[1] ; Ming Liu ^[1]
  1. [1] Qufu Normal University

     Qufu Normal University

     China

     
  2. [2] Guangdong Polytechnic Normal University

     Guangdong Polytechnic Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 3, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • A discrete Bazykin predator–prey model with predator intraspecific interactions and ratio-dependent functional response is proposed and investigated. A brief mathematical analysis of the model involves giving fixed points and analyzing local stability.

    Codimension-one bifurcations such as the transcritical, fold, flip, Neimark-Sacker bifurcations and codimension-two bifurcations, including the fold-flip bifurcation, 1:2, 1:3, and 1:4 strong resonances, are investigated. Accurate theoretical analysis and exquisite numerical simulation are given simultaneously, which support the main content of this manuscript. With the help of several local attraction basins, period, and Lyapunov exponent diagrams, an intriguing set of fascinating dynamics in the two-parameter spaces of integral step size with other parameters is uncovered. The results in this paper reveal that the dynamics of the discrete-time predator–prey system in both single-parameter and two-parameter spaces are inherently rich and complex.

• Referencias bibliográficas
  • 1. Williams, B.K., Nichols, J.D., Conroy, M.J.: Analysis and Management of Animal Populations. Academic Press, San Diego, CA (2002)
  • 2. Hembry, D.H., Hembry, M.G.: Ecological interactions and macroevolution: a new field with old roots. Annu. Rev. Ecol. Evol. Syst. 51, 215–243...
  • 3. Zhao, L.Z., Huang, C.D., Cao, J.D.: Dynamics of fractional-order predator-prey model incorporating two delays. Fractals 29, 2150014 (2021)
  • 4. Berryman, A.A.: The origins and evolutions of predator-prey theory. Ecology 73, 1530–1535 (1992)
  • 5. Ruan, S.G., Xiao, D.M.: Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61, 1445–1472...
  • 6. Zhou, H., Tang, B., Zhu, H.P., Tang, S.Y.: Bifurcation and dynamic analyses of non-monotonic predatorprey system with constant releasing...
  • 7. Xiao, D.M., Ruan, S.G.: Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response. J. Differ. Equ....
  • 8. Hu, D.P., Li, Y.Y., Liu, M., Bai, Y.Z.: Stability and Hopf bifurcation for a delayed predator-prey model with stage structure for prey...
  • 9. Mondal, N., Barman, D., Alam, S.: Impact of adult predator incited fear in a stage-structured preypredator model. Environ. Dev. Sustain....
  • 10. Li, S.M., Wang, X.L., Li, X.L., Wu, K.L.: Relaxation oscillations for Leslie-type predator-prey model with Holling type I response functional...
  • 11. Lu, M., Huang, J.C.: Global analysis in Bazykin’s model with Holling II functional response and predator competition. J. Differ. Equ....
  • 12. Arias, C.F., Blé, G., Falconi, M.: Dynamics of a discrete-time predator-prey system with Holling II functional response. Qual. Theory...
  • 13. Vishwakarma, K., Sen, M.: Influence of Allee effect in prey and hunting cooperation in predator with Holling type-III functional response....
  • 14. Huang, J.C., Ruan, S.G., Song, J.: Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional...
  • 15. Zhang, J., Su, J.: Bifurcations in a predator-prey model of Leslie-type with simplified Holling type IV functional response. Int. J. Bifurcat....
  • 16. Zhuo, X.L., Zhang, F.X.: Stability for a new discrete ratio-dependent predator-prey system. Qual. Theory Dyn. Syst. 17, 189–202 (2018)
  • 17. Dai, B.X., Zou, J.Z.: Periodic solutions of a discrete-time nonautonomous predator-prey system with the Beddington-DeAngelis functional...
  • 18. Li, X.Y., Wang, Q., Han, R.J.: An impulsive predator-prey system with modified Leslie-Gower functional response and diffusion. Qual. Theory...
  • 19. Gao, X.Y., Ishag, S.D., Fu, S.M., Li, W.J., Wang, W.M.: Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator-prey...
  • 20. Shang, Z.C., Qiao, Y.H.: Multiple bifurcations in a predator-prey system of modified Holling and Leslie type with double Allee effect...
  • 21. Li, Y.J., He, M.X., Li, Z.: Dynamics of a ratio-dependent Leslie-Gower predator-prey model with Allee effect and fear effect. Math. Comput....
  • 22. Gutierrez, A.P.: The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson’s blowflies as an...
  • 23. Cosner, C., Angelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on the functional response of predators. Theor. Popul....
  • 24. Arditi, R., Ginzburg, L.R., Akcakaya, H.R.: Variation in plankton densities among lakes: a case for ratio dependent predation models....
  • 25. Hanski, I.: The functional response of predator: worries about scale. Tree 6, 141–142 (1991)
  • 26. Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389–406 (1998)
  • 27. Haque, M.: Ratio-dependent predator-prey models of interacting populations. B. Math. Biol. 71, 430– 452 (2009)
  • 28. Banerjee, M., Abbas, S.: Existence and non-existence of spatial patterns in a ratio-dependent predatorprey model. Ecol. Complex. 21, 199–214...
  • 29. Arancibia-Ibarra, C., Aguirre, P., Flores, J., van Heijster, P.: Bifurcation analysis of a predator-prey model with predator intraspecific...
  • 30. Zhang, X.B., An, Q., Wang, L.: Spatiotemporal dynamics of a delayed diffusive ratio-dependent predator-prey model with fear effect. Nonlinear...
  • 31. Jiang, X., She, Z.K., Ruan, S.G.: Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional...
  • 32. Alekseev, V.V.: Effect of saturation factor on the dynamics of predator-prey system. Biofizika 18, 922–926 (1973)
  • 33. Bazykin, A.D.: Volterra system andMichaelis-Menten equation. In VoprosyMatematicheskoi Genetiki, Novosibirsk, pp. 103–143 (1974)
  • 34. Bazykin, A.D.: Structural and dynamical stability of model predator-prey systems. In: Int. Inst. Appl. Syst. Anal., Laxenburg, Austria...
  • 35. Bazykin, A.D., Berezovskaya, F.S., Buriev, T.I.: Dynamics of predator-prey system including predator saturation and competition. In: Faktory...
  • 36. Li, X.Y., Liu, Y.Q.: Transcritical bifurcation and flip bifurcation of a new discrete ratio-dependent predator-prey system. Qual. Theory...
  • 37. Singh, A., Deolia, P.: Dynamical analysis and chaos control in discrete-time prey-predator model. Commun. Nonlinear Sci. Numer. Simulat....
  • 38. Naik, P.A., Eskandari, Z., Yavuz, M., Zu, J.: Complex dynamics of a discrete-time BazykinBerezovskaya prey-predator model with a strong...
  • 39. Cheng, L.F., Cao, H.J.: Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee Effect. Commun. Nonlinear...
  • 40. Yu, Y., Cao, H.J.: Integral step size makes a difference to bifurcations of a discrete-time HindmarshRose model. Int. J. Bifurcat. Chaos...
  • 41. Hu, D.P., Cao, H.J.: Bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type. Commun. Nonlinear Sci....
  • 42. Zhang, L.M., Xu, Y.K., Liao, G.Y.: Codimension-two bifurcations and bifurcation controls in a discrete biological system with weak Allee...
  • 43. Liu, X.J., Liu, Y.: Codimension-two bifurcation analysis on a discrete Gierer-Meinhardt system. Int. J. Bifurcat. Chaos 30, 2050251 (2020)
  • 44. Strogatz, S.H.: Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering....
  • 45. Hastings, A., Hom, C.L., Ellner, S., Turchin, P., Godfray, H.C.J.: Chaos in ecology: Is mother nature a strange attractor? Annu. Rev....
  • 46. Maquet, J., Letellier, C., Aguirre, L.A.: Global models from the Canadian lynx cycles as a direct evidence for chaos in real ecosystems....
  • 47. Hossain, M., Garai, S., Jafari, S., Pal, N.: Bifurcation, chaos, multistability, and organized structures in a predator-prey model with...
  • 48. Chen, Q.L., Teng, Z.D., Wang, F.: Fold-flip and strong resonance bifurcations of a discrete-time mosquito model. Chaos Soliton. Fract....
  • 49. Abdelaziz, M.A.M., Ismail, A.I., Abdullah, F.A., Mohd, M.H.: Codimension one and two bifurcations of a discrete-time fractional-order...
  • 50. Naik, P.A., Eskandari, Z., Shahraki, H.E.: Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model. Math....
  • 51. Naik, P.A., Eskandari, Z., Avazzadeh, Z., Zu, J.: Multiple bifurcations of a discrete-time prey-predator model with mixed functional response....
  • 52. Barman, D., Roy, J., Alam, S.: Trade-off between fear level induced by predator and infection rate among prey species. J. Appl. Math....
  • 53. Barman, D., Kumar, V., Roy, J., Alam, S.: Modeling wind effect and herd behavior in a predator-prey system with spatiotemporal dynamics....
  • 54. Barman, D., Roy, J., Alam, S.: Impact of wind in the dynamics of prey-predator interactions. Math. Comput. Simulat. 191, 49–81 (2022)
  • 55. Wang, X.H., Wang, Z., Lu, J.W., Meng, B.: Stability, bifurcation and chaos of a discrete-time pair approximation epidemic model on adaptive...
  • 56. Kuznetsov, Yu.A.: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol. 112, 3rd edn. Springer, New York (2004)
  • 57. Govaerts, W., Kuznetsov, Y.A., Khoshsiar Ghaziani, R., Meijer, H.G.E.: Cl Matcontm: A Toolbox for Continuation and Bifurcation of Cycles...
  • 58. Kuznetsov, Yu.A., Meijer, H.G.E.: Numerical Bifurcation Analysis of Maps: From Theory to Software. Cambridge University Press, Cambridge...
  • 59. Bao, B.C., Chen, C.J., Bao, H., Zhang, X., Xu, Q., Chen, M.: Dynamical effects of neuron activation gradient on Hopfield neural network:...
  • 60. Rao, X.B., Chu, Y.D., Chang, Y.X., Zhang, J.G., Tian, Y.P.: Dynamics of a cracked rotor system with oil-film force in parameter space....
  • 61. Wang, F.J., Cao, H.J.: Model locking and quaiperiodicity in a discrete-time Chialvo neuron model. Commun. Nonlinear Sci. Numer. Simul....
  • 62. Liu, M., Meng, F.W., Hu, D.P.: Codimension-one and codimension-two bifurcations in a new discrete chaotic map based on gene regulatory...
  • 63. Yu, X., Liu, M., Zheng, Z.W., Hu, D.P.: Complex dynamics of a discrete-time SIR model with nonlinear incidence and recovery rates. Int....
  • 64. Blyuss, K.B., Kyrychko, S.N., Kyrychko, Y.N.: Time-delayed and stochastic effects in a predator-prey model with ratio dependence and Holling...