American Institute of Mathematical Sciences

Advanced
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 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.

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.

[1]

Mats Gyllenberg, Ping Yan. On the number of limit cycles for three dimensional Lotka-Volterra systems. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 347-352. doi: 10.3934/dcdsb.2009.11.347
[2]

Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
[3]

Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747
[4]

Ting-Hui Yang, Weinian Zhang, Kaijen Cheng. Global dynamics of three species omnivory models with Lotka-Volterra interaction. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2867-2881. doi: 10.3934/dcdsb.2016077
[5]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[6]

Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137
[7]

Bang-Sheng Han, Zhi-Cheng Wang, Zengji Du. Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1959-1983. doi: 10.3934/dcdsb.2020011
[8]

Guichen Lu, Zhengyi Lu. Permanence for two-species Lotka-Volterra cooperative systems with delays. Mathematical Biosciences & Engineering, 2008, 5 (3) : 477-484. doi: 10.3934/mbe.2008.5.477
[9]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043
[10]

Guo Lin, Wan-Tong Li, Shigui Ruan. Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 1-23. doi: 10.3934/dcds.2011.31.1
[11]

Dejun Fan, Xiaoyu Yi, Ling Xia, Jingliang Lv. Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2901-2922. doi: 10.3934/dcdsb.2018291
[12]

Meng Liu, Ke Wang. Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2495-2522. doi: 10.3934/dcds.2013.33.2495
[13]

Zhanyuan Hou, Stephen Baigent. Global stability and repulsion in autonomous Kolmogorov systems. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1205-1238. doi: 10.3934/cpaa.2015.14.1205
[14]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[15]

Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142
[16]

Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290
[17]

Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771
[18]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5141-5164. doi: 10.3934/dcds.2021071
[19]

Zecen He, Haihua Liang, Xiang Zhang. Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 421-441. doi: 10.3934/dcdsb.2021049
[20]

Chiun-Chuan Chen, Yin-Liang Huang, Li-Chang Hung, Chang-Hong Wu. Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats. Communications on Pure and Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy