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Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations
1. | School of Mathematical Sciences, Soochow University, Suzhou 215006, China, China, China |
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.
|
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. |
[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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