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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 and 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), 199-204. 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), 202-219. doi: 10.1017/S0334270000005300.

[3]

Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489. doi: 10.1006/jmaa.1994.1262.

[4]

I. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification, Biol. Cybern., 48 (1983), 201-211. 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), 1151-1160. 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), 779-815. doi: 10.1512/iumj.2005.54.2506.

[7]

J. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32 (1977), 82-95. 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), 103-117.

[9]

T. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type, in "World Congress of Nonlinear Analysts '92, Vo. I-IV" (ed. V. Lakshmikantham) (Tampa, FL, 1992), de Gruyter, Berlin, (1996), 395-406.

[10]

P. van den Driessche and M. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. 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), 689-701. 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), 135-145.

[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), 1623-1672. 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), 1-9. 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), 319-335. 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), 1445-1472.

[18]

H. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., 17 (1986), 1289-1318. 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), 580-610. 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), 2967-2974. 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), 420-439. 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), 1580-1597. doi: 10.1137/070702485.

[23]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290. 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), 494-510. doi: 10.1006/jdeq.2000.3982.

[25]

J. You, Perturbations of lower dimensional tori for Hamiltonian systems, J. Differential Equations, 152 (1999), 1-29. doi: 10.1006/jdeq.1998.3515.

[26]

X. Yuan, Construction of quasi-periodic breathers via KAM technique, Comm. Math. Phys., 226 (2002), 61-100. 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. 

show all references

References:
[1]

S. Ahmad, On the nonautonomous Lotka-Volterra competition equations, Proc. Amer. Math. Soc., 117 (1993), 199-204. 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), 202-219. doi: 10.1017/S0334270000005300.

[3]

Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489. doi: 10.1006/jmaa.1994.1262.

[4]

I. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification, Biol. Cybern., 48 (1983), 201-211. 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), 1151-1160. 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), 779-815. doi: 10.1512/iumj.2005.54.2506.

[7]

J. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32 (1977), 82-95. 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), 103-117.

[9]

T. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type, in "World Congress of Nonlinear Analysts '92, Vo. I-IV" (ed. V. Lakshmikantham) (Tampa, FL, 1992), de Gruyter, Berlin, (1996), 395-406.

[10]

P. van den Driessche and M. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. 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), 689-701. 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), 135-145.

[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), 1623-1672. 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), 1-9. 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), 319-335. 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), 1445-1472.

[18]

H. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., 17 (1986), 1289-1318. 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), 580-610. 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), 2967-2974. 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), 420-439. 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), 1580-1597. doi: 10.1137/070702485.

[23]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290. 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), 494-510. doi: 10.1006/jdeq.2000.3982.

[25]

J. You, Perturbations of lower dimensional tori for Hamiltonian systems, J. Differential Equations, 152 (1999), 1-29. doi: 10.1006/jdeq.1998.3515.

[26]

X. Yuan, Construction of quasi-periodic breathers via KAM technique, Comm. Math. Phys., 226 (2002), 61-100. 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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