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September  2013, 33(9): 4071-4093. doi: 10.3934/dcds.2013.33.4071

Heteroclinic limit cycles in competitive Kolmogorov systems

Zhanyuan Hou 1, and  Stephen Baigent 2,

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

Received  May 2012 Revised  January 2013 Published  March 2013

A notion of global attraction and repulsion of heteroclinic limit cycles is introduced for strongly competitive Kolmogorov systems. Conditions are obtained for the existence of cycles linking the full set of axial equilibria and their global asymptotic behaviour on the carrying simplex. The global dynamics of systems with a heteroclinic limit cycle is studied. Results are also obtained for Kolmogorov systems where some components vanish as $t\rightarrow \pm \infty$.
Keywords: Lotka-Volterra systems, Kolmogorov systems, global attractors, heteroclinic limit cycles., global repellers.
Mathematics Subject Classification: Primary: 34C37, 37C29; Secondary: 34D05, 34D45, 37C7.
Citation: Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Field and J. W. Swift, Stationary bifurcation to limit cycles and heteroclinic cycles, Nonlinearity, 4 (1991), 1001-1043.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative III: competing species, Nonlinearity, 1 (1988), 51-71.  Google Scholar

[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.  Google Scholar

[13]

J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mountains Math. Publ., 4 (1994), 105-116.  Google Scholar

[14]

J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," Cambridge University Press, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[21]

H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, UK, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[22]

H. Smith, "Monotone Dynamical Systems, An Introduction to Theory of Competitive and Cooperative Systems," Math. Surveys Monogr., 41, AMS, Providence, RI, 1995.  Google Scholar

[23]

Y. Takeuchi, "Global Dynamical Properties of Lotka-Volterra Systems," World Scentific, Singapore, 1996. doi: 10.1142/9789812830548.  Google Scholar

[24]

A. Tineo, Necessary and sufficient conditions for extinction of one species, Adv. Nonlinear Stud., 5 (2005), 57-71.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stab. Syst., 8 (1993), 189-217.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Field and J. W. Swift, Stationary bifurcation to limit cycles and heteroclinic cycles, Nonlinearity, 4 (1991), 1001-1043.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative III: competing species, Nonlinearity, 1 (1988), 51-71.  Google Scholar

[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.  Google Scholar

[13]

J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mountains Math. Publ., 4 (1994), 105-116.  Google Scholar

[14]

J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," Cambridge University Press, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[21]

H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, UK, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[22]

H. Smith, "Monotone Dynamical Systems, An Introduction to Theory of Competitive and Cooperative Systems," Math. Surveys Monogr., 41, AMS, Providence, RI, 1995.  Google Scholar

[23]

Y. Takeuchi, "Global Dynamical Properties of Lotka-Volterra Systems," World Scentific, Singapore, 1996. doi: 10.1142/9789812830548.  Google Scholar

[24]

A. Tineo, Necessary and sufficient conditions for extinction of one species, Adv. Nonlinear Stud., 5 (2005), 57-71.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stab. Syst., 8 (1993), 189-217.  Google Scholar

[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.  Google Scholar

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