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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

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.  Google Scholar

[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.  Google Scholar

[3]

Z. Amine and R. Ortega, A periodic prey-predator system,, J. Math. Anal. Appl., 185 (1994), 477.  doi: 10.1006/jmaa.1994.1262.  Google Scholar

[4]

I. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification,, Biol. Cybern., 48 (1983), 201.  doi: 10.1007/BF00318088.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82.  doi: 10.1137/0132006.  Google Scholar

[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.   Google Scholar

[9]

T. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type,, in, (1996), 395.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Gyllenberg, P. Yan and Y. Wang, Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems,, Physica D, 221 (2006), 135.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Lancaster, "Theory of Matrices," Academic Press,, New York-London, (1969).   Google Scholar

[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.  Google Scholar

[17]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response,, SIAM J. Appl. Math., 61 (2000), 1445.   Google Scholar

[18]

H. Smith, Periodic solutions of periodic competitive and cooperative systems,, SIAM J. Math. Anal., 17 (1986), 1289.  doi: 10.1137/0517091.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. You, Perturbations of lower dimensional tori for Hamiltonian systems,, J. Differential Equations, 152 (1999), 1.  doi: 10.1006/jdeq.1998.3515.  Google Scholar

[26]

X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61.  doi: 10.1007/s002200100593.  Google Scholar

[27]

X. Yuan and A. Nunes, A note on the reducibility of linear differential equations with quasiperiodic coefficients,, Int. J. Math. Sci., 2003 (): 4071.   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

Z. Amine and R. Ortega, A periodic prey-predator system,, J. Math. Anal. Appl., 185 (1994), 477.  doi: 10.1006/jmaa.1994.1262.  Google Scholar

[4]

I. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification,, Biol. Cybern., 48 (1983), 201.  doi: 10.1007/BF00318088.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82.  doi: 10.1137/0132006.  Google Scholar

[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.   Google Scholar

[9]

T. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type,, in, (1996), 395.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Gyllenberg, P. Yan and Y. Wang, Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems,, Physica D, 221 (2006), 135.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Lancaster, "Theory of Matrices," Academic Press,, New York-London, (1969).   Google Scholar

[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.  Google Scholar

[17]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response,, SIAM J. Appl. Math., 61 (2000), 1445.   Google Scholar

[18]

H. Smith, Periodic solutions of periodic competitive and cooperative systems,, SIAM J. Math. Anal., 17 (1986), 1289.  doi: 10.1137/0517091.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. You, Perturbations of lower dimensional tori for Hamiltonian systems,, J. Differential Equations, 152 (1999), 1.  doi: 10.1006/jdeq.1998.3515.  Google Scholar

[26]

X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61.  doi: 10.1007/s002200100593.  Google Scholar

[27]

X. Yuan and A. Nunes, A note on the reducibility of linear differential equations with quasiperiodic coefficients,, Int. J. Math. Sci., 2003 (): 4071.   Google Scholar

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