Abstract
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.
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References
Abbott, K.C., Dwyer, G.: Food limitation and insect outbreaks: complex dynamics in plant-herbivore models. J. Anim. Ecol. 76, 1004–1014 (2007)
Asfaw, M.D., Kassa, S.M., Lungu, E.M.: Coexistence thresholds in the dynamics of the plant-herbivore interaction with Allee effect and harvest. Int. J. Biomath. 11, 27 (2018)
Bravermana, E., Rodkina, A.: Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean. Comput. Math. Appl. 66, 2281–2294 (2013)
Caughley, G., Lawton, J.H.: Plant-herbivore systems. Theor. Ecol. 132–166 (1981)
Comins, H.N., McMurtrie, R.E.: Long-term response of nutrient-limited forests to CO002 enrichment, equilibrium behavior of plant-soil models. Ecol. Appl. 3(4), 666–681 (1993)
Din, Q., Shabbir, M.S., Asif, M., Ahmad, K.: Bifurcation analysis and chaos control for a plant-herbivore model with weak predator functional response. J. Biol. Dyn. 13, 481–501 (2019)
Din, Q.: Global behavior of a plant-herbivore model. Adv. Differ. Equ. 2015, 119 (2015). https://doi.org/10.1186/s13662-015-0458-y
Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer (2005)
Elsayed, E.M., Din, Q.: Period-doubling and Neimark-Sacker bifurcations of plant-herbivore models. Adv. Differ. Equ. 2019, 271 (2019). https://doi.org/10.1186/s13662-019-2200-7
Feng, Z., DeAngelis, D.L.: Mathematical Models of Plant-Herbivore Interactions. Chapman & Hall/CRC (2018)
Gotelli, N.J.: A Primer of Ecology. Sinauer Associates (2001)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
Hale, J. K., Kocak, J. K.: Dynamics and Bifurcations. Texts in Applied Mathematics, Springer-Verlag, New York (1991)
Hofbauer, J.: A general cooperation theorem for hypereycles. Mh. Math. 91, 233–240 (1981)
Hofbauer, J., Hutson, V., Jansen, W.: Coexistence for systems governed by difference equations of Lotka-Volterra type. J. Math. Biol. 25, 553–570 (1987)
Hutson, V.: A theorem on average Liapunov functions. Monatshefte für Mathematik 98, 267–275 (1984)
Hutson, V., Moran, W.: Persistence of species obeying difference equations. J. Math. Biol. 15, 203–213 (1982)
Jothi, S.S., Gunasekaran, M.: Chaos and bifurcation analysis of plant-herbivore system with intra-specific competitions. Int. J. Adv. Res. 3, 1359–1363 (2015)
Kalabušić, S., Drino, D., Pilav, E.: Global behavior and bifurcation in a class of host-parasitoid models with a constanthost refuge. Qual. Theory Dyn. Syst 19, 66 (2020). https://doi.org/10.1007/s12346-020-00403-3
Kalabušić, S., Drino, D., Pilav, E.: Period-doubling and Neimark-Sacker bifurcations of a Beddington host-parasitoid model with a host refuge effect. Int. J. Bifurc. Chaos 30(16) (2020). https://doi.org/10.1142/S0218127420502545
Kang, Y., Armbruster, D.: Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions. Int. J. Biomath. 04, 255–274 (2011)
Kang, Y., Armbruster, D., Kuang, Y.: Dynamics of a plant-herbivore model. J. Biol. Dyn. 2, 89–101 (2008)
Kapitaniak, T.: Chaos for Engineers: Theory, Applications & Control, 2nd edn. Springer-Verlag, New York (2000)
Kapitaniak, T.: Controlling Chaos: Theoretical and Practical Methods in Non-linear Dynamics. Academic Press (1996)
Kartal, S.: Dynamics of a plant-herbivore model with differential-difference equations. CogentMath 3 (2016). https://doi.org/10.1080/23311835.2015.1136198
Kon, R., Takeuchi, Y.: Permanence of host-parasitoid system. Nonlinear Anal. 47, 1383–1393 (2001)
Liu, R., Feng, Z., Zhu, H., DeAngelis, D.: Bifurcation analysis of a plant-herbivore model with toxin-determined functional response. J. Differ. Equ. 245, 442–467 (2008)
Lynch, S.: Dynamical Systems with Applications using Mathematica, 2nd edn. Birkha\(\ddot{u}\) (2010)
Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)
Ricker, W.E.: Stock and recruitment. J. Fish. Res. Board Can. 11, 559–623 (1954)
Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, USA (1995)
Saha, T., Bandyopadhyay, M.: Dynamical analysis of a plant-herbivore model: analysis of: bifurcation and global stability. J. Appl. Math. Comput. 19, 327–344 (2005)
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 model involving Allee effect and cannibalism. Adv. Differ. Equ. 2020, 379 (2020). https://doi.org/10.1186/s13662-020-02838-z
Turchin, P.: Complex Population Dynamics: A Theoretical/empirical Synthesis, vol. 35. Princeton University Press, Princeton (2003)
Ufuktepe, \(\rm \ddot{U}.\), Kapçak, S.: Applications of Discrete Dynamical Systems with Mathematica, Conference: RIMS vol. 1909 (2014)
Weiss, J.N., Garfinkel, A., Spano, M.L., Ditto, W.L.: Chaos and chaos control in biology. Clin. Invest 93, 1355–1360 (1994)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second Edition, Texts in Applied Mathematics, vol. 2. Springer-Verlag, New York (2003)
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Kalabušić, S., Pilav, E. Bifurcations, Permanence and Local Behavior of the Plant-Herbivore Model with Logistic Growth of Plant Biomass. Qual. Theory Dyn. Syst. 21, 26 (2022). https://doi.org/10.1007/s12346-022-00561-6
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DOI: https://doi.org/10.1007/s12346-022-00561-6