# American Institute of Mathematical Sciences

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

## 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

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 $t\rightarrow \pm \infty$.
Citation: Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete and 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. [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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##### 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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