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Heteroclinic limit cycles in competitive Kolmogorov systems
1. | School of Computing, London Metropolitan University, 166-220 Holloway Road, London N7 8DB, United Kingdom |
2. | Department of Mathematics, UCL, Gower Street, London WC1E 6BT, United Kingdom |
References:
[1] |
V. S. Afraimovich, M. I. Raminovich and P. Varona, Heteroclinic contours in neural ensembles and the winnerless competition principle, Internat. J. Bifur. Chaos, 14 (2004), 1195-1208.
doi: 10.1142/S0218127404009806. |
[2] |
P. Ashwin, O. Burylkob and Y. Maistrenko, Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators, Physica D, 237 (2008), 454-466.
doi: 10.1016/j.physd.2007.09.015. |
[3] |
S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems, Diff. Eq. Dyn. Syst., 20 (2012), 53-66.
doi: 10.1007/s12591-012-0103-0. |
[4] |
G. Butler and P. Waltman, Persistence in dynamical systems, J. Diff. Eq., 63 (1996), 255-263.
doi: 10.1016/0022-0396(86)90049-5. |
[5] |
C. Chi, S. Hsu and L. Wu, On the asymmetric May-Leonard model of three competing species, SIAM J. Appl. Math., 58 (1998), 211-226.
doi: 10.1137/S0036139994272060. |
[6] |
M. Corbera, J. Llibre and M. A. Teixeira, Symmetric periodic orbits near a heteroclinic loop in $R^3$ formed by two singular points, a semistable periodic orbit and their invariant manifolds, Physica D, 238 (2009), 699-705.
doi: 10.1016/j.physd.2009.01.002. |
[7] |
B. Feng, The heteroclinic cycle in the model of competition between $n$ species and its stability, Acta Mathematicae Applicatae Sinica, 14 (1998), 404-413.
doi: 10.1007/BF02683825. |
[8] |
M. Field and J. W. Swift, Stationary bifurcation to limit cycles and heteroclinic cycles, Nonlinearity, 4 (1991), 1001-1043. |
[9] |
M. Krupa and I. Melborne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergo. Th. & Dynam. Sys., 15 (1995), 121-147.
doi: 10.1017/S0143385700008270. |
[10] |
M. Krupa and I. Melborne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II, Proc. Roy. Soc. Edin., 134A (2004), 1177-1197.
doi: 10.1017/S0308210500003693. |
[11] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative III: competing species, Nonlinearity, 1 (1988), 51-71. |
[12] |
J. Hofbauer and K. Sigmund, On the stabilizing effect of predators and competitors on ecological communities, J. Math. Biol., 27 (1989), 537-548.
doi: 10.1007/BF00288433. |
[13] |
J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mountains Math. Publ., 4 (1994), 105-116. |
[14] |
J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," Cambridge University Press, New York, 1998. |
[15] |
Z. Hou, Vanishing components in autonomous competitive Lotka-Volterra systems, J. Math. Anal. Appl., 359 (2009), 302-310.
doi: 10.1016/j.jmaa.2009.05.054. |
[16] |
Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems, Dyn. Syst., 26 (2011), 367-390.
doi: 10.1080/14689367.2011.554384. |
[17] |
S.-B. Hsu and L.-I. W. Roeger, Heteroclinic cycles in the chemostat models and the winnerless competition principle, J. Math. Anal. Appl., 360 (2009), 599-608.
doi: 10.1016/j.jmaa.2009.07.006. |
[18] |
R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253. |
[19] |
K. Orihashi and Y. Aizawa, Global aspects of turbulence induced by heteroclinic cycles in competitive diffusion Lotka-Volterra equation, Physica D, 240 (2011), 1853-1862. |
[20] |
H. L. Smith, Periodic orbits of competitive and cooperative systems, J. Diff. Eq., 65 (1986), 361-373.
doi: 10.1016/0022-0396(86)90024-0. |
[21] |
H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, UK, 1995.
doi: 10.1017/CBO9780511530043. |
[22] |
H. Smith, "Monotone Dynamical Systems, An Introduction to Theory of Competitive and Cooperative Systems," Math. Surveys Monogr., 41, AMS, Providence, RI, 1995. |
[23] |
Y. Takeuchi, "Global Dynamical Properties of Lotka-Volterra Systems," World Scentific, Singapore, 1996.
doi: 10.1142/9789812830548. |
[24] |
A. Tineo, Necessary and sufficient conditions for extinction of one species, Adv. Nonlinear Stud., 5 (2005), 57-71. |
[25] |
D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Diff. Eq., 164 (2000), 1-15.
doi: 10.1006/jdeq.1999.3729. |
[26] |
E. C. Zeeman and M. L. Zeeman, An $n$-dimensional Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, 15 (2002), 2019-2032.
doi: 10.1088/0951-7715/15/6/312. |
[27] |
E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc., 355 (2003), 713-734.
doi: 10.1090/S0002-9947-02-03103-3. |
[28] |
M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stab. Syst., 8 (1993), 189-217. |
[29] |
X.-A. Zhang and L. Chen, The global dynamic behaviour of the competition model of three species, J. Math. Anal. and Appl., 245 (2000), 124-141.
doi: 10.1006/jmaa.2000.6742. |
show all references
References:
[1] |
V. S. Afraimovich, M. I. Raminovich and P. Varona, Heteroclinic contours in neural ensembles and the winnerless competition principle, Internat. J. Bifur. Chaos, 14 (2004), 1195-1208.
doi: 10.1142/S0218127404009806. |
[2] |
P. Ashwin, O. Burylkob and Y. Maistrenko, Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators, Physica D, 237 (2008), 454-466.
doi: 10.1016/j.physd.2007.09.015. |
[3] |
S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems, Diff. Eq. Dyn. Syst., 20 (2012), 53-66.
doi: 10.1007/s12591-012-0103-0. |
[4] |
G. Butler and P. Waltman, Persistence in dynamical systems, J. Diff. Eq., 63 (1996), 255-263.
doi: 10.1016/0022-0396(86)90049-5. |
[5] |
C. Chi, S. Hsu and L. Wu, On the asymmetric May-Leonard model of three competing species, SIAM J. Appl. Math., 58 (1998), 211-226.
doi: 10.1137/S0036139994272060. |
[6] |
M. Corbera, J. Llibre and M. A. Teixeira, Symmetric periodic orbits near a heteroclinic loop in $R^3$ formed by two singular points, a semistable periodic orbit and their invariant manifolds, Physica D, 238 (2009), 699-705.
doi: 10.1016/j.physd.2009.01.002. |
[7] |
B. Feng, The heteroclinic cycle in the model of competition between $n$ species and its stability, Acta Mathematicae Applicatae Sinica, 14 (1998), 404-413.
doi: 10.1007/BF02683825. |
[8] |
M. Field and J. W. Swift, Stationary bifurcation to limit cycles and heteroclinic cycles, Nonlinearity, 4 (1991), 1001-1043. |
[9] |
M. Krupa and I. Melborne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergo. Th. & Dynam. Sys., 15 (1995), 121-147.
doi: 10.1017/S0143385700008270. |
[10] |
M. Krupa and I. Melborne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II, Proc. Roy. Soc. Edin., 134A (2004), 1177-1197.
doi: 10.1017/S0308210500003693. |
[11] |
M. W. Hirsch, Systems of differential equations that are competitive or cooperative III: competing species, Nonlinearity, 1 (1988), 51-71. |
[12] |
J. Hofbauer and K. Sigmund, On the stabilizing effect of predators and competitors on ecological communities, J. Math. Biol., 27 (1989), 537-548.
doi: 10.1007/BF00288433. |
[13] |
J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mountains Math. Publ., 4 (1994), 105-116. |
[14] |
J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," Cambridge University Press, New York, 1998. |
[15] |
Z. Hou, Vanishing components in autonomous competitive Lotka-Volterra systems, J. Math. Anal. Appl., 359 (2009), 302-310.
doi: 10.1016/j.jmaa.2009.05.054. |
[16] |
Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems, Dyn. Syst., 26 (2011), 367-390.
doi: 10.1080/14689367.2011.554384. |
[17] |
S.-B. Hsu and L.-I. W. Roeger, Heteroclinic cycles in the chemostat models and the winnerless competition principle, J. Math. Anal. Appl., 360 (2009), 599-608.
doi: 10.1016/j.jmaa.2009.07.006. |
[18] |
R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253. |
[19] |
K. Orihashi and Y. Aizawa, Global aspects of turbulence induced by heteroclinic cycles in competitive diffusion Lotka-Volterra equation, Physica D, 240 (2011), 1853-1862. |
[20] |
H. L. Smith, Periodic orbits of competitive and cooperative systems, J. Diff. Eq., 65 (1986), 361-373.
doi: 10.1016/0022-0396(86)90024-0. |
[21] |
H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, UK, 1995.
doi: 10.1017/CBO9780511530043. |
[22] |
H. Smith, "Monotone Dynamical Systems, An Introduction to Theory of Competitive and Cooperative Systems," Math. Surveys Monogr., 41, AMS, Providence, RI, 1995. |
[23] |
Y. Takeuchi, "Global Dynamical Properties of Lotka-Volterra Systems," World Scentific, Singapore, 1996.
doi: 10.1142/9789812830548. |
[24] |
A. Tineo, Necessary and sufficient conditions for extinction of one species, Adv. Nonlinear Stud., 5 (2005), 57-71. |
[25] |
D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Diff. Eq., 164 (2000), 1-15.
doi: 10.1006/jdeq.1999.3729. |
[26] |
E. C. Zeeman and M. L. Zeeman, An $n$-dimensional Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, 15 (2002), 2019-2032.
doi: 10.1088/0951-7715/15/6/312. |
[27] |
E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc., 355 (2003), 713-734.
doi: 10.1090/S0002-9947-02-03103-3. |
[28] |
M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stab. Syst., 8 (1993), 189-217. |
[29] |
X.-A. Zhang and L. Chen, The global dynamic behaviour of the competition model of three species, J. Math. Anal. and Appl., 245 (2000), 124-141.
doi: 10.1006/jmaa.2000.6742. |
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