Dynamical Analysis of a Discrete Amensalism System with the Beddington–DeAngelis Functional Response and Allee Effect for the Unaffected Species

• Qimei Zhou ^[1] ; Fengde Chen ^[1]
  1. [1] Fuzhou University

     Fuzhou University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 1, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This research involves a discrete amensalism system with the Beddington–DeAngelis functional response and Allee effect for the unaffected species. We begin by investigating the presence and local stability of fixed points. Then, utilizing the central manifold theorem and bifurcation theory, we analyze a variety of codimension one and codimension two bifurcations, which include transcritical, pitchfork, fold, flip, fold-flip and 1 : 2 strong resonance bifurcations. These theoretical findings suggest that Allee effect serves a crucial role in stabilizing the population sizes of both species.

    In addition, Allee effect would make the system spend more time to achieve its stable steady-state solution. They are illustrated via numerical simulations.

• Referencias bibliográficas
  • 1. Gómez, J.M., González-Megías, A.: Asymmetrical interactions between ungulates and phytophagous insects: being different matters. Ecology....
  • 2. Ogada, D.L., Gadd, M.E., Ostfel, R.S., Young, T.P., Keesing, F.: Impacts of large herbivorous mammals on bird diversity and abundance in...
  • 3. Xi, X., Griffin, J.N., Sun, S.: Grasshoppers amensalistically suppress caterpillar performance and enhance plant biomass in an alpine meadow....
  • 4. García, C., Rendueles, M., Díaz, M.: Microbial amensalism in Lactobacillus casei and Pseudomonas taetrolens mixed culture. Bioproc. Biosyst....
  • 5. Sun, G.C.: Qualitative analysis on two populations amensalism model. J. Jiamusi Univ. (Nat. Sci. Ed.) 21(3), 283–286 (2003)
  • 6. Su, Q.Q., Chen, F.D.: The influence of partial closure for the populations to a non-selective harvesting Lotka–Volterra discrete amensalism...
  • 7. Guan, X.Y., Chen, F.D.: Dynamical analysis of a two species amensalism model with Beddington– DeAngelis functional response and Allee effect...
  • 8. Luo, D.M., Wang, Q.R.: Global dynamics of a Holling-II amensalism system with nonlinear growth rate and Allee effect on the first species....
  • 9. Xu, L.L., Xue, Y.L., Xie, X.D., Lin, Q.F.: Dynamic behaviors of an obligate commensal symbiosis model with Crowley–Martin functional responses....
  • 10. Liu, H.Y., Yu, H.G., Dai, C.J., Ma, Z.L., Wang, Q., Zhao, M.: Dynamical analysis of an aquatic amensalism model with non-selective harvesting...
  • 11. Wei, Z., Xia, Y.H., Zhang, T.H.: Stability and bifurcation analysis of an amensalism model with weak Allee effect. Qual. Theory Dyn. Syst....
  • 12. Zhao, M., Du, Y.F.: Stability and bifurcation analysis of an amensalism system with Allee effect. Adv. Diff. Equ. Article ID 341 (2020)
  • 13. Zhou, Q.M., Chen, F.D., Lin, S.J.: Complex dynamics analysis of a discrete amensalism system with a cover for the first species. Axioms....
  • 14. Chen, B.G.: Dynamic behaviors of a non-selective harvesting Lotka-Volterra amensalism model incorporating partial closure for the populations....
  • 15. Liu, Y., Zhao, L., Huang, X.Y, Deng, H.: Stability and bifurcation analysis of two species amensalism model with Michaelis-Menten type...
  • 16. Xiao, Z.W., Li, Z., Zhu, Z.L., Chen, F.D.: Hopf bifurcation and stability in a Beddington–DeAngelis predator-prey model with stage structure...
  • 17. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44, 331–340...
  • 18. De Angelis, D.L., Goldstein, R.A., O’neill, R.V.: A model for tropic interaction. Ecology. 56, 881–892 (1975)
  • 19. Skalski, G.T., Gilliam, J.F.: Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology....
  • 20. Zhang, L.M., Zou, L.: Bifurcations and control in a discrete predator-prey model with strong Allee effect. Internat. J. Bifur. Chaos....
  • 21. I¸sık, S.: Stability and period-doubling bifurcation in a modified commensal symbiosis model with Allee effect. Erzincan Univ. J. Sci....
  • 22. Pal, S., Sasmal, S.K., Pal, N.: Chaos control in a discrete-time predator-prey model with weak Allee effect. Int. J. Biomath. 11, 1850089...
  • 23. Chen, J.L., Chen, Y.M., Zhu, Z.L., Chen, F.D.: Stability and bifurcation of a discrete predator-prey system with Allee effect and other...
  • 24. Jiang, H., Rogers, T.D.: The discrete dynamics of symmetric competition in the plane. J. Math. Biol. 25, 573–596 (1987)
  • 25. Liu, X.L., Xiao, D.M.: Complex dynamic behaviors of a discrete-time predator-prey system. Chaos Solitons Fractals. 32, 80–94 (2007)
  • 26. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York...
  • 27. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton (1998)
  • 28. Luo, X.S., Chen, G., Wang, B.H., Fang, J.Q.: Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems....