Existence of periodic positive solutions to nonlinear Lotka-Volterra competition

• Autorzy: Benhadri Mimia , Caraballo Tomas , Zeghdoudi Halim

Identyfikatory

DOI

10.7494/OpMath.2020.40.3.341

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Krasnoselskii's fixed point theorem; positive periodic solutions; Lotka-Volterra competition systems; variable delays

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2020
• Tom: Vol. 40, no. 3
• Strony: 341--360
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Benhadri Mimia

• LAMAHIS Lab Faculty of Sciences Department of Mathematics University of Skikda P.O. Box 26, Skikda 21000, Algeria

autor

Caraballo Tomas

• Departamento de Ecuaciones Difererenciales y Analisis Numerico Universidad de Sevilla c/ Tarfia s/n, 41012 - Sevilla, Spain

autor

Zeghdoudi Halim

• LaPS Laboratory Badji-Mokhtar University P.O. Box 12, Annaba 23000, Algeria

Bibliografia

• [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. Denes, 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 ol Qualitative Theory ol 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.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).