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Modelling the Dynamics in a Predator–Prey System with Allee Effects and Anti-predator Behavior

  • Tao Wen [1] ; Yancong Xu [1] ; Mu He [2] ; Libin Rong [3]
    1. [1] China Jiliang University

      China Jiliang University

      China

    2. [2] Xi’an Jiaotong-Liverpool University

      Xi’an Jiaotong-Liverpool University

      China

    3. [3] University of Florida

      University of Florida

      Estados Unidos

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

    • This paper studies a predator–prey model with strong and weak Allee effects and anti-predator behavior using a dynamical system approach. We perform a detailed bifurcation analysis including saddle-node bifurcation, Hopf bifurcation of codimension 3, cusp of codimension 3, cusp type Bogdanov–Takens bifurcation of codimension 3, and codimension-2 cusp of the limit cycle. The involvement of strong and weak Allee effects provides a new regime shift mechanism, which indicates the transition from a homoclinic cycle to a new heteroclinic cycle connecting two boundary equilibria induced by the Allee effect and the carrying capacity. The role of anti-predator behavior is fully uncovered by studying the interaction with the Allee effect. It is the first time that we find a codimension-2 cusp of infinitesimal limit cycle in the predator– prey system, which indicates the existence of a coexistence region of three limit cycles due to the weak Allee effect. Different from the scenario in the reference (Aguirre et al.

      in SIAM J Appl Math 69(5):1244–1262, 2009), it is a new generating mechanism of limit cycle bifurcating from one Hopf bifurcation point with two saddle-node bifurcation points on the limit cycle branch, and the double limit cycle curve originates from a codimension-2 degenerate Hopf bifurcation point and disappears at another one.

      The dynamics of the model with the Allee effect and anti-predator behavior are shown to be more complicated than those for other predator–prey systems. The biological interpretations of the bifurcation diagram and phase portrait are also provided.

  • Referencias bibliográficas
    • 1. Lotka, A.J.: The Elements of Physical Biology. Williams & Wilkins (1925)
    • 2. Volterra, V.: Fluctuations in the abundance of species considered mathematically. Nature 119(2983), 12–13 (1927)
    • 3. Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10(6),...
    • 4. Zhu, H., Campbell, S.A., Wolkowicz, G.S.K.: Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM...
    • 5. Freedman, B.: Deterministic mathematical models in population ecology. Biometrics 22(7), 219–236 (1980)
    • 6. Gause, G.F.: Experimental analysis of Vito Volterra’s mathematical theory of the struggle for existence. Science 79(2036), 16–17 (1934)
    • 7. Foster, W.A., Treherne, J.E.: Evidence for the dilution effect in the selfish herd from fish predation on a marine insect. Nature 293(5832),...
    • 8. Huang, J., Xiao, D.: Analyses of bifurcations and stability in a predator-prey system with Holling type-IV functional response. Acta Math....
    • 9. Myers, R.A., Worm, B.: Rapid worldwide depletion of large predatory fish communities. Nature 423, 280–283 (2003)
    • 10. Freedman, H.I., Wolkowicz, G.S.K.: Predator–prey systems with group defense: the paradox of enrochment revisted. Bull. Math. Biol. 8,...
    • 11. Xiao, D., Ruan, S.: Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61, 1445–1472...
    • 12. Rothe, F., Shafer, D.S.: Multiple bifurcation in a predator-prey system with nonmonotonic predator response. Proc. R. Soc. Edinb. 120,...
    • 13. Naik, P.A., Eskandari, Z., Yavuz, M., Zu, J.: Complex dynamics of a discrete-time Bazykin–Berezovskaya prey-predator model with a strong...
    • 14. Eskandari, Z., Avazzadeh, Z., Ghaziani, R.K.: Theoretical and numerical bifurcation analysis of a predator–prey system with ratio-dependence....
    • 15. Li, B., Eskandari, Z., Avazzadeh, Z.: Strong resonance bifurcations for a discrete-time prey–predator model. J. Comput. Appl. Math. (2023)....
    • 16. Eskandari, Z., Avazzadeh, Z., Ghaziani, R.K., Li, B.: Dynamics and bifurcations of a discrete-time Lotka–Volterra model using nonstandard...
    • 17. Du, Y., Niu, B., Guo, Y., Wei, J.: Double Hopf bifurcation in delayed reaction–diffusion systems. J. Dyn. Differ. Equ. 32, 313–358 (2020)
    • 18. Duan, D., Niu, B., Wei, J.: Hopf–Hopf bifurcation and chaotic attractors in a delayed diffusive predator–prey model with fear effect....
    • 19. Allee, W.C.: Animal Aggregations: A Study in General Sociology. The University of Chicago Press (1931)
    • 20. Allee, W.C.: An ecological audit. Ecology 20(3), 418–421 (1939)
    • 21. Arsie, A., Kottegoda, C., Shan, C.: A predator–prey system with generalized Holling type IV functional response and Allee effects in prey....
    • 22. Johnson, D.M., Liebhold, A.M., Tobin, P.C., et al.: Allee effects and pulsed invasion by the gypsy moth. Nature 444(7117), 361–363 (2006)
    • 23. Kent, A., Doncaster, C.P., Sluckin, T.: Consequences for predators of rescue and Allee effects on prey. Ecol. Model. 162(3), 233–245 (2003)
    • 24. Wang, M.H., Kot, M.: Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 171(1), 83–97 (2001)
    • 25. Dai, G., Tang, M.: Coexistence region and global dynamics of a harvested predator–prey system. SIAM J. Appl. Math. 51(1), 193–210 (1998)
    • 26. Pauly, D., et al.: Towards sustainability in world fisheries. Nature 418, 689–695 (2002)
    • 27. González-Olivares, E., Meneses-Alcay, H., et al.: Multiple stability and uniqueness of limit cycle in a Gause-type predator-prey model...
    • 28. Cai, L., Chen, G., Xiao, D.: Multiparametric bifurcations of an epidemiological model with strong Allee effect. J. Math. Biol. 67(2),...
    • 29. Garain, K., Mandal, P.S.: Bubbling and hydra effect in a population system with Allee effect. Ecol. Complex. 47, 1–14 (2021)
    • 30. Mendez, V., Sans, C., Lopis, I., Campos, D.: Extinction conditions for isolated populations with Allee effect. Math. Biosci. 232, 78–86...
    • 31. Rocha, J.L., Fournier-Prunaret, D., Taha, Abdel-Kaddous.: Strong and weak Allee effects and chaotic dynamics in Richards’ growths. Discrete...
    • 32. Rocha, J.L., Taha, Abdel-Kaddous., Fournier-Prunaret, D.: Big bang bifurcation in von Bertalanffy’s dynamics with strong and weak Allee...
    • 33. Boukal, D.S., Berec, L.: Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. J. Theor. Biol....
    • 34. Brassil, C.E.: Mean time to extinction of a metapopulation with an Allee effect. Ecol. Model. 143, 9–16 (2001)
    • 35. Schreiber, S.J.: Allee effects, extinctions, and chaotic transients in simple population models. Theor. Popul. Biol. 64(2), 201–209 (2003)
    • 36. Aguirre, P., González-Olivares, E., Sáez, E.: Three limit cycles in a Leslie–Gower predator–prey model with additive Allee effect. SIAM...
    • 37. Li, C., Rousseau, C.: A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: the cusp...
    • 38. Tang, B., Xiao, Y.N.: Bifurcation analysis of a predator–prey model with anti-predator behaviour. Chaos Solitons Fractals 70, 58–68 (2015)
    • 39. Huang, J., Gong, Y., Ruan, S.: Bifurcation analysis in a predator–prey model with constant-yield predator harvesting. Discrete Contin....
    • 40. Leard, B., Lewis, C., Rebaza, J.: Dynamics of ratio-dependent predator-prey models with nonconstant harvesting. Discrete Contin. Dyn....
    • 41. Yang, Y., Meng, F.W., Xu, Y.C.: Global bifurcation analysis in a predator-prey system with simplified Holling IV functional response and...
    • 42. Xiao, D., Zhou, Y.: Qualitative analysis of an epidemic model. Can. Appl. Math. Q. 14(4), 480–484 (2006)
    • 43. Perko, L.: Differential Equations and Dynamical Systems. Springer (2001)
    • 44. Lamontagne, Y., Coutu, C., Rousseau, C.: Bifurcation analysis of a predator-prey system with generalised Holling type III functional response....
    • 45. Shan, C., Zhu, H.: Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds. J. Differ. Equ. 257(5),...
    • 46. Dumortier, F., Roussarie, R., Sotomayor, J.: Generic 3-parameter families of vector fields on the plane, unfolding a singularity with...
    • 47. Chow, S.N., Li, C., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (1994)
    • 48. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York (1983)
    • 49. Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Theory of Bifurcation of Dynamic Systems on a Plane, Israel Program for Science...
    • 50. Golubitsky, M., Langford, W.F.: Classification and unfolding of degenerate Hopf bifurcation. J. Differ. Equ. 41, 375–415 (1981)
    • 51. Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equations. Transl. from the Chinese by Anthony Wing-Kwok...
    • 52. Witte, V.D., Rossa, F.D., Govaerts, W., Kuznetsov, Y.A.: Numerical periodic normalization for codim 2 bifurcations of limit cycles: computational...
    • 53. Iooss, G.: Global characterization of the normal form for a vector field near a closed orbit. J. Differ. Equ. 76, 47–76 (1988)
    • 54. Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X., Zhang,...
    • 55. Govaerts, W., Kuznetsov, Yu.A., Dhooge, A.: Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM J. Sci. Comput. 27,...
    • 56. Xu, Y.C., Wei, L.J., Jiang, X.Y., Zhu, Z.R.: Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete...
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