Forwards Attractor Structures in a Planar Cooperative Non-autonomous Lotka–Volterra System

• Autores: Juan García Fuentes, Piotr Kalita, José Antonio Langa Rosado
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
• Idioma: inglés
• Enlaces
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• Resumen

  • The global attractor of a dissipative dynamical system provides the necessary information to understand the asymptotic dynamics of all the system’s solutions. A crucial question consists in finding the structure of this set. In this paper we provide a full characterization of the structure of attractors for a planar non-autonomous Lotka–Volterra cooperative system. We show sufficient conditions for the existence of forward attractors and give a full description of them by proving the existence of such bounded global solutions that all bounded global solutions join them, i.e. converge towards them when time tends to plus and minus infinity. These results generalize those known in an autonomous framework. The case of particular interest in our work is the situation where globally forward-stable non-autonomous solutions have both coordinates strictly positive. We study this case in detail and obtain sufficient conditions that the problem parameters must satisfy in order to obtain various structures of nonautonomous attractors. This allows us to understand different paths of the solutions towards the unique globally stable one.

• Referencias bibliográficas
  • 1. Ahmad, S., Lazer, A.: On the nonautonomous n-competing species problems. Appl. Anal. 57, 309–323 (1995)
  • 2. Ahmad, S., Lazer, A.: Necessary and sufficient average growth in a Lotka–Volterra system. Nonlinear Anal. 34, 191–228 (1998)
  • 3. Ahmad, S., Lazer, A.: Average conditions for global asymptotic stability in an nonautonomous Lotka– Volterra system. Nonlinear Anal. 40,...
  • 4. Buonomo, B., Chitnis, N., d’Onofrio, A.: Seasonality in epidemic models: a literature review. Ricerche Mat. 67, 7–25 (2018)
  • 5. Bortolan, M.C., Carvalho, A.N., Langa, J.A.: Attractors Under Autonomous and Non-autonomous Perturbations. Mathematical Surveys and Monographs,...
  • 6. Battelli, F., Palmer, K.J.: Criteria for exponential dichotomy for triangular systems. J. Math. Anal. Appl. 428, 525–543 (2015)
  • 7. Bai, Z., Zhou, Y.: Existence of two periodic solutions for a non-autonomous sir epidemic model. Appl. Math. Model. 35(1), 382–391 (2011)
  • 8. Bai, Z., Zhou, Y., Zhang, T.: Existence of multiple periodic solutions for an sir model with seasonality. Nonlinear Anal. Theory Methods...
  • 9. Carvalho, A.N., Langa, J.A., Robinson, J.C.: Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems. Applied Mathematical...
  • 10. Garcia-Fuentes, J., Langa, J.A., Kalita, P., Suárez, A.: Forwards attractors for non-autonomous Lotka– Volterra cooperative systems: a...
  • 11. Guerrero, G., Langa, J.A., Suarez, A.: Architecture of attractor determines dynamics on mutualistic complex networks. Nonlinear Anal....
  • 12. Gopalsamy, K.: Global asymptotic stability in a periodic Lotka–Volterra system. ANZIAM J. 27, 66–72 (1986)
  • 13. Gopalsamy, K.: Global asymptotic stability in an almost periodic Lotka–Volterra system. ANZIAM J. 27, 346–360 (1986)
  • 14. Heesterbeek, J.: A brief history of R0 and a recipe for its calculation. Acta Biotheor. 50, 189–204 (2002)
  • 15. Kuznetsov, Y., Muratori, S., Rinaldi, S.: Bifurcations and chaos in a periodic predator–prey model. Int. J. Bifurc. Chaos 2(1), 15–25...
  • 16. Kloeden, P., Rasmussen, M.: Nonautonomous Dynamical Systems. Mathematical Surveys and Monographs, vol. 176. American Mathematical Society,...
  • 17. Labouriau, I.S., Rodrigues, A.A.P.: Bifurcations from an attracting heteroclinic cycle under periodic forcing. J. Differ. Equ. 269(5),...
  • 18. Portillo, J.R., Soler-Toscano, F., Langa, J.A.: Global structural stability and the role of cooperation in mutualistic systems. PLoS ONE...
  • 19. Redheffer, R.: Nonautonomous Lotka–Volterra systems. I. J. Differ. Equ. 127, 519–541 (1996)
  • 20. Rinaldi, S., Muratori, S., Kuznetsov, Y.: Multiple attractors, catastrophes and chaos in seasonally perturbed predator–prey communities....
  • 21. Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Americal Mathematical Society, Providence (2011)
  • 22. Takeuchi, Y.: Global Asymptotic Dynamical Properties of Lotka–Volterra Systems. World Scientific Publishing, Singapore (1996)
  • 23. Tsai, T.-L., Dawes, J.H.P.: Dynamics near a periodically-perturbed robust heteroclinic cycle. Physica D Nonlinear Phenom. 262, 14–34 (2013)
  • 24. Thieme, H.R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166(2), 173–201 (2000)
  • 25. Tineo, A.: Necessary and sufficient conditions for extinction of one species. Adv. Nonlinear Stud. 5, 57–71 (2005)
  • 26. Wesley, C.L., Allen, L.J.S.: The basic reproduction number in epidemic models with periodic demographics. J. Biol. Dyn. 3(2–3), 116–129...