Complete Global Stability of a Metapopulation Model and Its Dynamically Consistent Discrete Models

• Dang, Quang A ^[1] ; Manh Tuan Hoang ^[1]
  1. [1] Vietnam Academy of Science and Technology

     Vietnam Academy of Science and Technology

     Vietnam

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 18, Nº 2, 2019, págs. 461-475
• Idioma: inglés
• DOI: 10.1007/s12346-018-0295-y
• Enlaces
  • Texto completo
• Resumen

  • In this paper we establish the complete global stability of a metapopulation model based on Lyapunov direct method, and the Poincare–Bendixson theorem in combination with the Bendixson–Dulac criterion. Besides, we construct nonstandard finite difference (NSFD) schemes preserving essential properties of the metapopulation model such as positivity, boundedness and monotone convergence of the solutions, equilibria and their stability. The numerical simulations confirm the validity of the obtained theoretical results and the advantages of NSFD schemes over standard finite difference schemes.

• Referencias bibliográficas
  • 1. Allen, L.J.S.: An Introduction to Mathematical Biology. Prentice Hall, New Jersey (2007)
  • 2. Amarasekare, P., Possingham, H.: Patch dynamics and metapopulation theory: the case of successional species. J. Theor. Biol. 209, 333–344...
  • 3. Dang, Q.A., Hoang, M.T.: Dynamically consistent discrete metapopulation model. J. Differ. Equ. Appl. 22, 1325–1349 (2016)
  • 4. Dang, Q.A., Tuan, H.M.: Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation...
  • 5. Dimitrov, D.T., Kojouharov, H.V.: Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems. Appl....
  • 6. Dimitrov, D.T., Kojouharov, H.V.: Stability-preserving finite-difference methods for general multidimensional autonomous dynamical systems....
  • 7. Dimitrov, D.T., Kojouharov, H.V.: Nonstandard finite-difference methods for predator-prey models with general functional response. Math....
  • 8. Georgescu, P., Zhang, H.: Lyapunov functionals for two-species mutualisms. Appl. Math. Comput. 226, 754–764 (2014)
  • 9. Iggidr, A., Bensoubaya, M.: New results on the stability of discrete-time systems and applications to control problems. J. Math. Anal....
  • 10. Khalil, H.K.: Nonlinear Systems. Prentice Hall, New Jersey (2002)
  • 11. Korobeinikov, A.: A Lyapunov function for Leslie-Gower predator–prey models. Appl. Math. Lett. 14, 697–699 (2001)
  • 12. Korobeinikov, A., Wake, G.C.: Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Appl. Math. Lett....
  • 13. Korobeinikov, A.: Global properties of basic virus dynamics models. Bull. Math. Biol. 66, 879–883 (2004)
  • 14. Korobeinikov, A.: Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages. Bull. Math. Biol. 1, 75–83...
  • 15. Korobeinikov, A.: Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 1871–1886 (2007)
  • 16. Korobeinikov, A.: Stability of ecosystem: global properties of a general predator-prey model. Math. Med. Biol. 26, 309–321 (2009)
  • 17. La Salle, J., Lefschetz, S.: Stability by Liapunov’s Direct Method. Academic Press, New York (1961)
  • 18. Lyapunov, A.M.: The General Problem of the Stability of Motion. Taylor & Francis, London (1992)
  • 19. Martcheva, M.: An Introduction to Mathematical Epidemiology. Springer, New York (2015)
  • 20. Mickens, R.E.: Applications of Nonstandard Finite Difference Schemes. World Scientific, Singapore (2000)
  • 21. Mickens, R.E.: Nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 9, 823–847 (2005)
  • 22. Thieme, H.R.: Convergence results and a Poincar Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol....
  • 23. Vargas De Leon, C.: On the global stability of SIS, SIR and SIRS epidemic models with standard incidence. Chaos Solitons Fractals 44,...
  • 24. Vargas De Leon, C.: Lyapunov functions for two-species cooperative systems. Appl. Math. Comput. 219, 2493–2497 (2012)
  • 25. Yang, W., Li, X.: Global asymptotical stability of a n + 1 dimensional Leslie-Gower predator-prey model. Appl. Math. Comput. 235,...