Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations

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July 2012, 17(5): 1537-1550. doi: 10.3934/dcdsb.2012.17.1537

Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations

Qihuai Liu ^1, , Dingbian Qian ^1, and Zhiguo Wang ^1,

1.	School of Mathematical Sciences, Soochow University, Suzhou 215006, China, China, China

Received November 2010 Revised January 2012 Published March 2012

Abstract
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In this paper, we prove the existence of positive quasi-periodic solutions for the Lotka-Volterra competition systems with quasi-periodic coefficients by KAM technique. The result shows that, in most case, quasi-periodic solutions exist for sufficiently small quasi-periodic perturbations of the autonomous Lotka-Volterra systems. Moreover, these quasi-periodic solutions will tend to an equilibrium of the autonomous Lotka-Volterra systems.

Keywords: KAM iteration, quasi-periodic solution, small denominator., Lotka-Volterra competition system.

Mathematics Subject Classification: Primary: 37N25, 92D25, 34A30, 34C2.

Citation: Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537

References:

[1]	S. Ahmad, On the nonautonomous Lotka-Volterra competition equations,, Proc. Amer. Math. Soc., 117 (1993), 199. doi: 10.1090/S0002-9939-1993-1143013-3.
[2]	C. Alvarez and A. Lazer, An application of topological degree to the periodic competing species problem,, J. Austral. Math. Soc. Ser. B, 28 (1986), 202. doi: 10.1017/S0334270000005300.
[3]	Z. Amine and R. Ortega, A periodic prey-predator system,, J. Math. Anal. Appl., 185 (1994), 477. doi: 10.1006/jmaa.1994.1262.
[4]	I. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification,, Biol. Cybern., 48 (1983), 201. doi: 10.1007/BF00318088.
[5]	H. Z. Cong, L. F. Mi and X. P. Yuan, Positive quasi-periodic solutions to Lotka-Volterra system,, Sci. China Math., 53 (2010), 1151. doi: 10.1007/s11425-009-0217-1.
[6]	M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems,, Indiana Univ. Math. J., 54 (2005), 779. doi: 10.1512/iumj.2005.54.2506.
[7]	J. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82. doi: 10.1137/0132006.
[8]	T. Ding, H. Huang and F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations,, Discrete Continuous Dynam. Systems, 1 (1995), 103.
[9]	T. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type,, in, (1996), 395.
[10]	P. van den Driessche and M. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits,, SIAM J. Appl. Math., 58 (1998), 227. doi: 10.1137/S0036139995294767.
[11]	H. Freedman and J. Wu, Periodic solutions of single-species models with periodic delay,, SIAM J. Math. Anal., 23 (1992), 689. doi: 10.1137/0523035.
[12]	M. Gyllenberg, P. Yan and Y. Wang, Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems,, Physica D, 221 (2006), 135.
[13]	J. Jiang, J. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding,, J. Differential Equations, 246 (2009), 1623. doi: 10.1016/j.jde.2008.10.008.
[14]	A. R. Hausrath and R. F. Manásevich, Periodic solutions of a periodically perturbed Lotka-Volterra equation using the Poincaré-Birkhoff theorem,, J. Math. Anal. Appl., 157 (1991), 1. doi: 10.1016/0022-247X(91)90132-J.
[15]	P. Lancaster, "Theory of Matrices," Academic Press,, New York-London, (1969).
[16]	P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach,, J. Mathe. Biol., 11 (1981), 319. doi: 10.1007/BF00276900.
[17]	S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response,, SIAM J. Appl. Math., 61 (2000), 1445.
[18]	H. Smith, Periodic solutions of periodic competitive and cooperative systems,, SIAM J. Math. Anal., 17 (1986), 1289. doi: 10.1137/0517091.
[19]	X. Tang, D. Cao and X. Zou, Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay,, J. Differential Equations, 228 (2006), 580. doi: 10.1016/j.jde.2006.06.007.
[20]	X. Tang and X. Zou, On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments,, Proc. Amer. Math. Soc., 134 (2006), 2967. doi: 10.1090/S0002-9939-06-08320-1.
[21]	X. Tang and X. Zou, 3/2-type criteria for global attractivity of Lotka-Volterra competition system without instantaneous negative feedbacks,, J. Differential Equations, 186 (2002), 420. doi: 10.1016/S0022-0396(02)00011-6.
[22]	Y. Xia and M. Han, New conditions on the existence and stability of periodic solution in Lotka-Volterra's population system,, SIAM J. Appl. Math., 69 (2009), 1580. doi: 10.1137/070702485.
[23]	D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system,, J. Math. Biol., 43 (2001), 268. doi: 10.1007/s002850100097.
[24]	D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response,, J. Differential Equations, 176 (2001), 494. doi: 10.1006/jdeq.2000.3982.
[25]	J. You, Perturbations of lower dimensional tori for Hamiltonian systems,, J. Differential Equations, 152 (1999), 1. doi: 10.1006/jdeq.1998.3515.
[26]	X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61. doi: 10.1007/s002200100593.
[27]	X. Yuan and A. Nunes, A note on the reducibility of linear differential equations with quasiperiodic coefficients,, Int. J. Math. Sci., 2003 (): 4071.

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References:

[1]	S. Ahmad, On the nonautonomous Lotka-Volterra competition equations,, Proc. Amer. Math. Soc., 117 (1993), 199. doi: 10.1090/S0002-9939-1993-1143013-3.
[2]	C. Alvarez and A. Lazer, An application of topological degree to the periodic competing species problem,, J. Austral. Math. Soc. Ser. B, 28 (1986), 202. doi: 10.1017/S0334270000005300.
[3]	Z. Amine and R. Ortega, A periodic prey-predator system,, J. Math. Anal. Appl., 185 (1994), 477. doi: 10.1006/jmaa.1994.1262.
[4]	I. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification,, Biol. Cybern., 48 (1983), 201. doi: 10.1007/BF00318088.
[5]	H. Z. Cong, L. F. Mi and X. P. Yuan, Positive quasi-periodic solutions to Lotka-Volterra system,, Sci. China Math., 53 (2010), 1151. doi: 10.1007/s11425-009-0217-1.
[6]	M. Conti, S. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems,, Indiana Univ. Math. J., 54 (2005), 779. doi: 10.1512/iumj.2005.54.2506.
[7]	J. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82. doi: 10.1137/0132006.
[8]	T. Ding, H. Huang and F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations,, Discrete Continuous Dynam. Systems, 1 (1995), 103.
[9]	T. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type,, in, (1996), 395.
[10]	P. van den Driessche and M. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits,, SIAM J. Appl. Math., 58 (1998), 227. doi: 10.1137/S0036139995294767.
[11]	H. Freedman and J. Wu, Periodic solutions of single-species models with periodic delay,, SIAM J. Math. Anal., 23 (1992), 689. doi: 10.1137/0523035.
[12]	M. Gyllenberg, P. Yan and Y. Wang, Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems,, Physica D, 221 (2006), 135.
[13]	J. Jiang, J. Mierczyński and Y. Wang, Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding,, J. Differential Equations, 246 (2009), 1623. doi: 10.1016/j.jde.2008.10.008.
[14]	A. R. Hausrath and R. F. Manásevich, Periodic solutions of a periodically perturbed Lotka-Volterra equation using the Poincaré-Birkhoff theorem,, J. Math. Anal. Appl., 157 (1991), 1. doi: 10.1016/0022-247X(91)90132-J.
[15]	P. Lancaster, "Theory of Matrices," Academic Press,, New York-London, (1969).
[16]	P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach,, J. Mathe. Biol., 11 (1981), 319. doi: 10.1007/BF00276900.
[17]	S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response,, SIAM J. Appl. Math., 61 (2000), 1445.
[18]	H. Smith, Periodic solutions of periodic competitive and cooperative systems,, SIAM J. Math. Anal., 17 (1986), 1289. doi: 10.1137/0517091.
[19]	X. Tang, D. Cao and X. Zou, Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay,, J. Differential Equations, 228 (2006), 580. doi: 10.1016/j.jde.2006.06.007.
[20]	X. Tang and X. Zou, On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments,, Proc. Amer. Math. Soc., 134 (2006), 2967. doi: 10.1090/S0002-9939-06-08320-1.
[21]	X. Tang and X. Zou, 3/2-type criteria for global attractivity of Lotka-Volterra competition system without instantaneous negative feedbacks,, J. Differential Equations, 186 (2002), 420. doi: 10.1016/S0022-0396(02)00011-6.
[22]	Y. Xia and M. Han, New conditions on the existence and stability of periodic solution in Lotka-Volterra's population system,, SIAM J. Appl. Math., 69 (2009), 1580. doi: 10.1137/070702485.
[23]	D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system,, J. Math. Biol., 43 (2001), 268. doi: 10.1007/s002850100097.
[24]	D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response,, J. Differential Equations, 176 (2001), 494. doi: 10.1006/jdeq.2000.3982.
[25]	J. You, Perturbations of lower dimensional tori for Hamiltonian systems,, J. Differential Equations, 152 (1999), 1. doi: 10.1006/jdeq.1998.3515.
[26]	X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61. doi: 10.1007/s002200100593.
[27]	X. Yuan and A. Nunes, A note on the reducibility of linear differential equations with quasiperiodic coefficients,, Int. J. Math. Sci., 2003 (): 4071.

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