Opuscula Math. 40, no. 3 (2020), 341-360
https://doi.org/10.7494/OpMath.2020.40.3.341

 
Opuscula Mathematica

Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systems

Mimia Benhadri
Tomás Caraballo
Halim Zeghdoudi

Abstract. We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature.

Keywords: Krasnoselskii's fixed point theorem, positive periodic solutions, Lotka-Volterra competition systems, variable delays.

Mathematics Subject Classification: 34K20, 34K13, 92B20.

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  1. S. Ahmad, On the nonautonomous Lotka-Volterra competition equations, Proc. Amer. Math. Soc. 117 (1993), 199-204.
  2. C. Alvarez, A.C. Lazer, An application of topological degree to the periodic competing species model, J. Aust. Math. Soc. Ser. B 28 (1986), 202-219.
  3. A. Battaaz, F. Zanolin, Coexistence states for periodic competition Kolmogorov systems, J. Math. Anal. Appl. 219 (1998), 179-199.
  4. F.D. Chen, Periodic solution and almost periodic solution for a delay multispecies logarithmic population model, Appl. Math. Comput. 171 (2005), 760-770.
  5. F.D. Chen, Periodic solutions and almost periodic solutions of a neutral multispecies logarithmic population model, Appl. Math. Comput. 176 (2006), 431-441.
  6. S. Chen, T. Wang, J. Zhang, Positive periodic solution for non-autonomous competition Lotka-Volterra patch system with time delay, Nonlinear Anal. Real World Appl. 5 (2004), 409-419.
  7. T. Cheon, Evolutionary stability of ecological hierarchy, Physical Review Letters 90 (2003) 25, Article ID 258105, 4 pages.
  8. A. Dénes, L. Hatvani, On the asymptotic behaviour of solutions of an asymptotically Lotka-Volterra model, Electron. J. Qual. Theory Differ. Equ. 67 (2016), 1-10.
  9. M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpn Ayala competition model, Comput. Math. Appl. 40 (2000), 1141-1151.
  10. M. Fan, K. Wang, D.Q. Jiang, Existence and global attractivity of positive peridic solutions of n-species Lotka-Volterra competition systems with several deviating arguments, Math. Biosci. 160 (1999), 47-61.
  11. P. Gao, Hamiltonian structure and first integrals for the Lotka-Volterra systems, Physics Letters A 273 (2000) 1-2, 85-96.
  12. K. Geisshirt, E. Praestgaard, S. Toxvaerd, Oscillating chemical reactions and phase separation simulated by molecular dynamics, J. Chem. Phys. 107 (1997) 22, 9406-9412.
  13. S.A.H. Geritz, M. Gyllenberg, Seven answers from adaptive dynamics, J. Evol. Biol. 18 (2005), 1174-1177.
  14. K. Gopalsamy, Global asymptotical stability in a periodic Lotka-Volterra system, J. Aust. Math. Soc. Ser. B 29 (1985), 66-72.
  15. M. Gyllenberg, Y. Wang, Dynamics of the periodic type-K competitive Kolmogorov systems, J. Differ. Equ. 205 (2004), 50-76.
  16. D. Hu, Z. Zhang, Four positive periodic solutions to a Lotka-Volterra cooperative system with harvesting terms, Nonlinear Anal. Real World Appl. 11 (2010), 1115-1121.
  17. M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
  18. Y.K. Li, On a periodic delay logistic type population model, Ann. Differential Equations 14 (1998), 29-36.
  19. Y.K. Li, Periodic solutions for delay Lotka-Volterra competition systems, J. Math. Anal. Appl. 246 (2000), 230-244.
  20. G. Lin, Y. Hong, Periodic solution in nonautonomous predator-prey system with delays, Nonlinear Anal. Real World Appl. 10 (2009), 1589-1600.
  21. A.J. Lotka, Undamped oscillations derived from the law of mass action, J. Am. Chem. Soc. 42 (1920), 1595-1599.
  22. S. Lu, On the existence of positive periodic solutions to a Lotka-Volterra cooperative population model with multiple delays, Nonlinear Anal. 68 (2008), 1746-1753.
  23. X. Lv, S.P. Lu, P. Yan, Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments, Nonlinear Anal. Real World Appl. 11 (2010), 574-583.
  24. A. Provata, G.A. Tsekouras, Spontaneous formation of dynamical patterns with fractal fronts in the cyclic lattice Lotka-Volterra model, Physical Review E 67 (2003) 5, part 2, Article ID 056602.
  25. Y.R. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, Electronic Journal of Qualitative Theory of Differential Equations 16 (2007), 1-10.
  26. H.L. Royden, Real Analysis, MacMillan Publishing Company, New York, 1998.
  27. X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974.
  28. V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, [in:] R.N. Chapman (ed.), Animal Ecology, McGraw-Hill, New York, 1926.
  29. G. Zhang, S.S. Cheng, Positive periodic solutions of coupled delay differential systems depending on two parameters, Taiwan. Math. J. 8 (2004), 639-652.
  30. H.Y. Zhao, L. Sun, Periodic oscillatory and global attractivity for chemostat model involving distributed delays, Nonlinear Anal. Real World Appl. 7 (2006), 385-394.
  • Tomás Caraballo (corresponding author)
  • ORCID iD https://orcid.org/0000-0003-4697-898X
  • Departamento de Ecuaciones Difererenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 - Sevilla, Spain
  • Communicated by Josef Diblík.
  • Received: 2018-12-23.
  • Revised: 2020-02-23.
  • Accepted: 2020-02-29.
  • Published online: 2020-04-04.
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Cite this article as:
Mimia Benhadri, Tomás Caraballo, Halim Zeghdoudi, Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systems, Opuscula Math. 40, no. 3 (2020), 341-360, https://doi.org/10.7494/OpMath.2020.40.3.341

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