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Global Harvesting and Stocking Dynamics in a Modified Rosenzweig–MacArthur Model

  • Yue Yang [1] ; Yancong Xu [1] ; Fanwei Meng [2] ; Libin Rong [3]
    1. [1] China Jiliang University

      China Jiliang University

      China

    2. [2] Qufu Normal University

      Qufu Normal University

      China

    3. [3] University of Florida

      University of Florida

      Estados Unidos

    Mostrar afiliaciones +
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
  • Idioma: inglés
  • DOI: 10.1007/s12346-024-01056-2
  • Enlaces
  • Resumen

    • Effective management of predator–prey systems is crucial for sustaining ecological balance and preserving biodiversity, which requires full understanding the dynamics of such systems with harvesting and stocking. This paper aims to investigate the global dynamics of a Rosenzweig–MacArthur model considering the interplay of these intervention practices. We reveal that this model undergoes a sequence of bifurcations, including cusp of codimensions 2 and 3, saddle-node bifurcation, Bogdanov–Takens (BT) bifurcation of codimensions 2 and 3, and degenerate Hopf bifurcation of codimension 2. In particular, a codimension-2 cusp of limit cycles is found, which indicates the coexistence of three limit cycles. An interesting and novel scenario is discovered:

      two distinct homoclinic cycle curves connect their respective BT bifurcation points.

      This differs from most models where a single homoclinic cycle curve may connect both BT bifurcation points. Moreover, we find that two families of limit cycles converge toward a heteroclinic cycle, signaling the risk of overexploitation. From a biological perspective, the prey population may undergo extinction for all initial states under large constant harvesting rate. Further, the simultaneous stocking of both populations is not conducive to the coexistence of both species; the stocking of one population and the harvesting of the other will promote the coexistence of two populations; while the simultaneous harvesting of two populations may result in multiple limit cycles, which effectively underscore the positive effect of harvesting and stocking. Identifying the optimal timing to harvest or stock predators and prey is crucial to prevent system collapse. This work promotes to a deeper understanding of the dynamics of ecosystems when harvesting and stocking occurs simultaneously. Further, it reveals the important roles of harvesting and stocking, contributing to the effective management of predator–prey systems.

  • Referencias bibliográficas
    • 1. Lotka, A.J.: Elements of Physical Biology. Williams & Wilkins, Baltimore (1925)
    • 2. Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558– 560 (1926). https://doi.org/10.1038/118558a0
    • 3. Leslie, P.H., Gower, J.C.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika...
    • 4. Shigesada, N., Kawasaki, K.: Biological invasions: theory and practice. Japan. J. Ecol. 1:1 (1997). https://doi.org/10.18960/seitai.47.3_339...
    • 5. Murray, J.D.: Mathematical Biology: I. An Introduction. Interdisciplinary Applied Mathematics. Springer, Berlin (2002)
    • 6. Li, B.T., Kuang, Y.: Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predatorprey system. SIAM J. Appl. Math. 67,...
    • 7. Li, Y.L., Xiao, D.M.: Bifurcations of a predator-prey system of Holling and Leslie types. Chaos Soliton. Fract. 34, 606–620 (2007). https://doi.org/10.1016/j.chaos.2006.03.068
    • 8. Huang, W.Z.: Traveling wave solutions for a class of predator-prey systems. J. Dyn. Differ. Equ. 24, 633–644 (2012). https://doi.org/10.1007/s10884-012-9255-4
    • 9. Zhu, C.R., Kong, L.: Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete and Cont....
    • 10. Seo, G., Kot, M.: A comparison of two predator-prey models with Holling’s type I functional response. Math. Biosci. 212, 161–179 (2008)....
    • 11. Sarif, N., Sarwardi, S.: Analysis of Bogdanov–Takens bifurcation of codimension 2 in a Gause-type model with constant harvesting of both...
    • 12. Wen, T., Xu, Y.C., He, M., Rong, L.B.: Modelling the dynamics in a predator-prey system with Allee effects and anti-predator behavior....
    • 13. May, R.M., Beddington, J.R., Clark, C.W., Holt, S.J., Laws, R.M.: Management of multispecies fisheries. Science 205, 267–277 (1979). https://doi.org/10.1126/science.205.4403.267
    • 14. Brauer, F., Soudack, A.C.: Coexistence properties of some predator-prey systems under constant rate harvesting and stocking. J. Math....
    • 15. Li, C., Rousseau, C.: A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: the cusp...
    • 16. Dai, G.R., Tang, M.X.: Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 58, 193–210 (1998)....
    • 17. Xiao, D.M., Jennings, L.S.: Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting. SIAM J. Appl. Math....
    • 18. Etoua, R.M., Rousseau, C.: Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function...
    • 19. Laurin, S., Rousseau, C.: Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling...
    • 20. Brauer, F., Soudack, A.C.: Stability regions in predator-prey systems with constant-rate prey harvesting. J. Math. Biol. 8, 55–71 (1979)....
    • 21. Xiao, D.M., Ruan, S.G.: Bogdanov–Takens bifurcations in predator-prey systems with constant rate harvesting. Fields Inst. Commun. 21,...
    • 22. Brauer, F., Soudack, A.C.: Stability regions and transition phenomena for harvested predator-prey systems. J. Math. Biol. 7, 319–337 (1979)....
    • 23. Brauer, F., Soudack, A.C.: Constant-rate stocking of predator-prey systems. J. Math. Biol. 11, 1–14 (1981). https://doi.org/10.1007/BF00275820
    • 24. Myerscough, M.R., Gray, B.F., Hogarth, W.L., Norbury, J.: An analysis of an ordinary differential equation model for a two-species predator-prey...
    • 25. Hogarth, W.L., Norbury, J., Cunning, I., Sommers, K.: Stability of a predator-prey model with harvesting. Ecol. Model. 62, 83–106 (1992)....
    • 26. Peng, G.J., Jiang, Y.L., Li, C.P.: Bifurcations of a Holling-type II predator-prey system with constant rate harvesting. Int. J. Bifurcat....
    • 27. Ruan, S.G., Xiao, D.M.: Imperfect and Bogdanov–Takens bifurcations in biological models: from harvesting of species to isolation of infectives....
    • 28. Lin, X.Q., Xu, Y.C., Gao, D.Z., Fan, G.H.: Bifurcation and overexploitation in Rosenzweig–Macarthur model. Discrete Contin. Dyn. Syst....
    • 29. Hsu, S.B.: On global stability of a predator-prey system. Math. Biosci. 39, 1–10 (1978). https://doi. org/10.1016/0025-5564(78)90025-1
    • 30. Shan, C.H., Zhu, H.P.: Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds. J. Differ. Equ....
    • 31. Lamontagne, Y., Coutu, C., Rousseau, C.: Bifurcation analysis of a predator-prey system with generalised Holling type III functional response....
    • 32. Bogdanov, R.I.: Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues. Funct. Anal. Appl....
    • 33. Bogdanov, R.I.: Bifurcation of the limit cycle of a family of plane vector fields/versal deformations of a singularity of a vector field...
    • 34. Takens, F.: Forced oscillations and bifurcations. In: Global Analysis of Dynamical Systems, pp. 11–71. CRC Press, Cambridge (2001)
    • 35. Perko, L.: Differential equations and dynamical systems. Differ. Equat. Dyn. Sys. 7, 181–314 (2001). https://doi.org/10.1007/978-1-4613-0003-8_3
    • 36. Li, C.Z., Li, J.Q., Ma, Z.E.: Codimension 3 BT bifurcations in an epidemic model with a nonlinear incidence. Discrete Cont. Dyn. Sys....
    • 37. Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, B., Paffenroth, R.C., Sandstede, B., Wang, X.J.,...
    • 38. Xu, Y.C., Yang, Y., Meng, F.W., Ruan, S.G.: Degenerate codimension-2 cusp of limit cycles in a Holling–Tanner model with harvesting and...
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