A Discrete‑Time Model for Consumer–Resource Interaction with Stability, Bifurcation and Chaos Control

• Din, Qamar ^[1] ; Khan, Muhammad Irfan ^[1]
  1. [1] University of Poonch Rawalakot
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00488-4
• Enlaces
  • Texto completo (pdf)
• Resumen

  • Keeping in mind the interactions between budmoths and the quality of larch trees located in the Swiss Alps (a mountain range in Switzerland), a discrete-time model is proposed and studied. The novel model is proposed with implementation of a nonlinear functional response that incorporates plant quality. The proposed functional response is validated with real observed data of larch budmoth interactions.

    Furthermore, we investigate the qualitative behavior of the proposed discrete-time system with interactions between budmoths and the quality of larch trees. Proofs of the boundedness of solutions, and the existence of fxed points and their topological classifcation are carried out. It is proved that the system experiences perioddoubling bifurcation at its positive fxed point using the center manifold theorem and normal forms theory. Moreover, existence and direction for the torus bifurcation are also investigated for larch budmoth interactions. Bifurcating and fuctuating behaviors of the system are controlled through utilization of chaos control strategies.

    Numerical simulations are presented to demonstrate the theoretical fndings. At the end, theoretical investigations are validated with feld and experimental data.

• Referencias bibliográficas
  • 1. Ginzburg, L.R., Taneyhill, D.E.: Population cycles of forest Lepidoptera: a maternal efect hypothesis. J. Anim. Ecol. 63, 79–92 (1994)
  • 2. Myers, J.H.: Can a general hypothesis explain population cycles of forest Lepidoptera? Adv. Ecol. Res. 18, 179–242 (1988)
  • 3. Baltensweiler, W., Rubli, D.: Dispersal: an important driving force of the cyclic population dynamics of the larch bud moth. For. Snow...
  • 4. Wermelinger, B., Forster, B., Nievergelt, D.: Cycles and importance of the larch budmoth. WSL Fact Sheet 61, 1–12 (2018)
  • 5. Zimmer, C.: Life after chaos. Science 284, 83–86 (1999)
  • 6. Esper, J., Buntgen, U., Frank, D.. C., Nievergelt, D., Liebhold, A.: 1200 years of regular outbreaks in alpine insects. Proc. R. Soc. Lond....
  • 7. Konter, O., Esper, J., Liebhold, A., Kyncl, T., Schneider, L., Duthorn, E., Buntgen, U.: Tree-ring evidence for the historical absence...
  • 8. Baltensweiler, W., Fischlin, A.: The larch budmoth in the Alps. In: Berryman, A. (ed.) Dynamics of Forest Insect Populations: Patterns,...
  • 9. Baltensweiler, W.: Why the larch bud-moth cycle collapsed in the subalpine larch-cembran pine forests in the year 1990 for the frst time...
  • 10. Berryman, A.A.: What causes population cycles of forest Lepidoptera? Trends. Ecol. Evol. 11, 28–32 (1996)
  • 11. Battipaglia, G., et al.: Long-term efects of climate and land-use change on larch budmoth outbreaks in the French Alps. Clim. Res. 62,...
  • 12. Turchin, P.: Complex Population Dynamics. Princeton University Press, New Jersey (2003)
  • 13. Jang, S.R.-J., Johnson, D.M.: Dynamics of discrete-time larch budmoth population models. J. Biol. Dyn. 3, 209–223 (2009)
  • 14. Jang, S.R.-J., Yu, J.-L.: Models of plant quality and larch budmoth interaction. Nonlinear Anal. Theory Methods Appl. 71(12), e1904–e1908...
  • 15. De Silva, T.M.M., Jang, S.R.-J.: Period-doubling and Neimark–Sacker bifurcations in a larch budmoth population model. J. Difer. Equ. Appl....
  • 16. Iyengar, S.V., Balakrishnan, J., Kurths, J.: Impact of climate change on larch budmoth cyclic outbreaks. Sci. Rep. 6, 27845 (2016)
  • 17. Balakrishnan, J., Iyengar, S.V., Kurths, J.: Missing cycles: efect of climate change on population dynamics. Indian Acad. Sci. Conf. Ser....
  • 18. Ali, I., Saeed, U., Din, Q.: Bifurcation analysis and chaos control in a discrete-time plant quality and larch budmoth interaction model...
  • 19. Yang, X.: Uniform persistence and periodic solutions for a discrete predatorprey system with delays. J. Math. Anal. Appl. 316, 161–177...
  • 20. Carr, J.: Application of Center Manifold Theory. Springer, New York (1981)
  • 21. Guzowska, M., Luís, R., Elaydi, S.: Bifurcation and invariant manifolds of the logistic competition model. J. Difer. Equ. Appl. 17(12),...
  • 22. Luís, R., Elaydi, S., Oliveira, H.: Stability of a Ricker-type competition model and the competitive exclusion principle. J. Biol. Dyn....
  • 23. Karydas, N., Schinas, J.: The center manifold theorem for a discrete system. Appl. Anal. 44(3–4), 267–284 (1992)
  • 24. Kulenović, M.R.S., Merino, O.: Invariant manifolds for competitive discrete systems in the plane. Int. J. Bifurc. Chaos 20(8), 2471–2486...
  • 25. Psarros, N., Papaschinopoulos, G., Schinas, C.J.: Semistability of two systems of diference equations using centre manifold theory. Math....
  • 26. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
  • 27. Robinson, C.: Dynamical Systems: Stability. Symbolic Dynamics and Chaos, Boca Raton, New York (1999)
  • 28. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)
  • 29. Wan, Y.H.: Computation of the stability condition for the Hopf bifurcation of difeomorphism on R2. SIAM. J. Appl. Math. 34, 167–175 (1978)
  • 30. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1997)
  • 31. Din, Q.: Stability, bifurcation analysis and chaos control for a predator-prey system. J. Vib. Control 25(3), 612–626 (2019)
  • 32. Din, Q., Shabbir, M.S., Khan, M.A., Ahmad, K.: Bifurcation analysis and chaos control for a plantherbivore model with weak predator functional...
  • 33. Abbasi, M.. A., Din, Q.: Under the infuence of crowding efects: stability, bifurcation and chaos control for a discrete-time predator–prey...
  • 34. Din, Q., Hussain, M.: Controlling chaos and Neimark–Sacker bifurcation in a host-parasitoid model. Asian J. Control 21(3), 1202–1215 (2019)
  • 35. Din, Q., Iqbal, M.A.: Bifurcation analysis and chaos control for a discrete-time enzyme model. Z. Naturforsch. A 74(1), 1–14 (2019)
  • 36. Ishaque, W., Din, Q., Taj, M., Iqbal, M.A.: Bifurcation and chaos control in a discrete-time predatorprey model with nonlinear saturated...
  • 37. Elsayed, E.M., Din, Q.: Period-doubling and Neimark–Sacker bifurcations of plant-herbivore models. Adv. Difer. Equ. 2019, 271 (2019)
  • 38. Din, Q.: A novel chaos control strategy for discrete-time Brusselator models. J. Math. Chem. 56(10), 3045–3075 (2018)
  • 39. Din, Q.: Bifurcation analysis and chaos control in discrete-time glycolysis models. J. Math. Chem. 56(3), 904–931 (2018)
  • 40. Din, Q., Donchev, T., Kolev, D.: Stability, bifurcation analysis and chaos control in chlorine dioxideiodine-malonic acid reaction. MATCH...
  • 41. Din, Q., Saeed, U.: Bifurcation analysis and chaos control in a host-parasitoid model. Math. Method Appl. Sci. 40(14), 5391–5406 (2017)
  • 42. Din, Q.: Qualitative analysis and chaos control in a density-dependent host-parasitoid system. Int. J. Dyn. Control 6(2), 778–798 (2018)
  • 43. Din, Q.: Neimark–Sacker bifurcation and chaos control in Hassell–Varley model. J. Difer. Equ. Appl. 23(4), 741–762 (2017)
  • 44. Din, Q., Haider, K.: Discretization, bifurcation analysis and chaos control for Schnakenberg model. J. Math. Chem. 58(8), 1615–1649 (2020)
  • 45. Din, Q., Saleem, N., Shabbir, M.S.: A class of discrete predator–prey interaction with bifurcation analysis and chaos control. Math. Model....
  • 46. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)
  • 47. Luo, X.S., Chen, G.R., Wang, B.H., Fang, J.Q.: Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical...
  • 48. Baltensweiler, W., Weber, U.M., Cherubini, P.: Tracing the infuence of larch-bud-moth insect outbreaks and weather conditions on larch...