Bifurcations, Permanence and Local Behavior of the Plant-Herbivore Model with Logistic Growth of Plant Biomass

• Kalabušic, S. ^[1] ; Pilav, E. ^[1]
  1. [1] University of Sarajevo

     University of Sarajevo

     Bosnia y Herzegovina

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 2, 2022
• Idioma: inglés
• DOI: 10.1007/s12346-022-00561-6
• Enlaces
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• Resumen

  • This paper investigates the plant-herbivore model’s dynamics. The plant’s biomass without herbivores growth with logistic equation assuming that the herbivore (parasitization) occurs after the host’s density-dependent growth regulation occurs. We give a topological classification of the equilibrium points. We show that the boundary equilibrium undergoes the transcritical, fold, and period-doubling bifurcation, whereas the interior equilibrium undergoes a Neimark-Sacker bifurcation. We use the OGY method to control chaos produced by period-doubling bifurcation. The system exhibits bistability between the stable interior attractors in the interior and the stable attractors in the x−boundary logistic dynamics (periodic orbits and strange attractors) for particular numerical values of parameters. Sufficient conditions for the permanence of the plant-herbivores system are obtained, ensuring the coexistence of both species.

• Referencias bibliográficas
  • 1. Abbott, K.C., Dwyer, G.: Food limitation and insect outbreaks: complex dynamics in plant-herbivore models. J. Anim. Ecol. 76, 1004–1014...
  • 2. Asfaw, M.D., Kassa, S.M., Lungu, E.M.: Coexistence thresholds in the dynamics of the plant-herbivore interaction with Allee effect and...
  • 3. Bravermana, E., Rodkina, A.: Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean....
  • 4. Caughley, G., Lawton, J.H.: Plant-herbivore systems. Theor. Ecol. 132–166 (1981)
  • 5. Comins, H.N., McMurtrie, R.E.: Long-term response of nutrient-limited forests to CO002 enrichment, equilibrium behavior of plant-soil models....
  • 6. Din, Q., Shabbir, M.S., Asif, M., Ahmad, K.: Bifurcation analysis and chaos control for a plantherbivore model with weak predator functional...
  • 7. Din, Q.: Global behavior of a plant-herbivore model. Adv. Differ. Equ. 2015, 119 (2015). https://doi. org/10.1186/s13662-015-0458-y
  • 8. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer (2005)
  • 9. Elsayed, E.M., Din, Q.: Period-doubling and Neimark-Sacker bifurcations of plant-herbivore models. Adv. Differ. Equ. 2019, 271 (2019)....
  • 10. Feng, Z., DeAngelis, D.L.: Mathematical Models of Plant-Herbivore Interactions. Chapman & Hall/CRC (2018)
  • 11. Gotelli, N.J.: A Primer of Ecology. Sinauer Associates (2001)
  • 12. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
  • 13. Hale, J. K., Kocak, J. K.: Dynamics and Bifurcations. Texts in Applied Mathematics, Springer-Verlag, New York (1991)
  • 14. Hofbauer, J.: A general cooperation theorem for hypereycles. Mh. Math. 91, 233–240 (1981)
  • 15. Hofbauer, J., Hutson, V., Jansen, W.: Coexistence for systems governed by difference equations of Lotka-Volterra type. J. Math. Biol....
  • 16. Hutson, V.: A theorem on average Liapunov functions. Monatshefte für Mathematik 98, 267–275 (1984)
  • 17. Hutson, V., Moran, W.: Persistence of species obeying difference equations. J. Math. Biol. 15, 203–213 (1982)
  • 18. Jothi, S.S., Gunasekaran, M.: Chaos and bifurcation analysis of plant-herbivore system with intraspecific competitions. Int. J. Adv. Res....
  • 19. Kalabuši´c, S., Drino, D., Pilav, E.: Global behavior and bifurcation in a class of host-parasitoid models with a constanthost refuge....
  • 20. Kalabuši´c, S., Drino, D., Pilav, E.: Period-doubling and Neimark-Sacker bifurcations of a Beddington host-parasitoid model with a host...
  • 21. Kang, Y., Armbruster, D.: Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions....
  • 22. Kang, Y., Armbruster, D., Kuang, Y.: Dynamics of a plant-herbivore model. J. Biol. Dyn. 2, 89–101 (2008)
  • 23. Kapitaniak, T.: Chaos for Engineers: Theory, Applications & Control, 2nd edn. Springer-Verlag, New York (2000)
  • 24. Kapitaniak, T.: Controlling Chaos: Theoretical and Practical Methods in Non-linear Dynamics. Academic Press (1996)
  • 25. Kartal, S.: Dynamics of a plant-herbivore model with differential-difference equations. CogentMath 3 (2016). https://doi.org/10.1080/23311835.2015.1136198
  • 26. Kon, R., Takeuchi, Y.: Permanence of host-parasitoid system. Nonlinear Anal. 47, 1383–1393 (2001)
  • 27. Liu, R., Feng, Z., Zhu, H., DeAngelis, D.: Bifurcation analysis of a plant-herbivore model with toxindetermined functional response. J....
  • 28. Lynch, S.: Dynamical Systems with Applications using Mathematica, 2nd edn. Birkhau¨ (2010)
  • 29. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)
  • 30. Ricker, W.E.: Stock and recruitment. J. Fish. Res. Board Can. 11, 559–623 (1954)
  • 31. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, USA (1995)
  • 32. Saha, T., Bandyopadhyay, M.: Dynamical analysis of a plant-herbivore model: analysis of: bifurcation and global stability. J. Appl. Math....
  • 33. Shabbir,M.S., Din, Q., Ahmad, K., Tassaddiq, A., Hassan Soori, A., Khan,Mu.A.: Stability, bifurcation, and chaos control of a novel discrete-time...
  • 34. Turchin, P.: Complex Population Dynamics: A Theoretical/empirical Synthesis, vol. 35. Princeton University Press, Princeton (2003)
  • 35. Ufuktepe, UR., Kapçak, S.: Applications of Discrete Dynamical Systems with Mathematica, Conference: RIMS vol. 1909 (2014)
  • 36. Weiss, J.N., Garfinkel, A., Spano, M.L., Ditto, W.L.: Chaos and chaos control in biology. Clin. Invest 93, 1355–1360 (1994)
  • 37. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second Edition, Texts in Applied Mathematics, vol. 2. Springer-Verlag,...