May  2015, 14(3): 1205-1238. doi: 10.3934/cpaa.2015.14.1205

Global stability and repulsion in autonomous Kolmogorov systems

1. 

School of Computing, London Metropolitan University, 166-220 Holloway Road, London N7 8DB

2. 

Department of Mathematics, UCL, Gower Street, London WC1E 6BT

Received  September 2014 Revised  December 2014 Published  March 2015

Criteria are established for the global attraction, or global repulsion on a compact invariant set, of interior and boundary fixed points of Kolmogorov systems. In particular, the notions of diagonal stability and Split Lyapunov stability that have found wide success for Lotka-Volterra systems are extended for Kolmogorov systems. Several examples from theoretical ecology and evolutionary game theory are discussed to illustrate the results.
Citation: Zhanyuan Hou, Stephen Baigent. Global stability and repulsion in autonomous Kolmogorov systems. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1205-1238. doi: 10.3934/cpaa.2015.14.1205
References:
[1]

S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems,, \emph{Differential Equations Dynam. Systems}, 20 (2012), 53. doi: 10.1007/s12591-012-0103-0. Google Scholar

[2]

R. M. Goodwin, Chaotic Economic Dynamics,, Clarendon Press, (1990). Google Scholar

[3]

A. S. Hacinliyan, I. Kusbeyzi and O. O. Aybar, Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior,, \emph{Nonlinear Dynam.}, 62 (2010), 17. doi: 10.1007/s11071-010-9695-5. Google Scholar

[4]

R. Haygood, Coexistence in MacArthur-style consumer-resource models,, \emph{Theor. Popul. Biol.}, 61 (2002), 215. Google Scholar

[5]

A. Hastings, Global stability of two species systems,, \emph{J. Math. Biol.}, 5 (1978), 399. doi: 10.1007/BF00276109. Google Scholar

[6]

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

[7]

M. W. Hirsch and H. L. Smith, Monotone dynamical systems,, in \emph{Handbook of Differential Equations: Ordinary Differential Equations 2}, (2006), 1. Google Scholar

[8]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems,, Cambridge University Press, (1998). Google Scholar

[9]

J. Hofbauer, W. H. Sandholm, Stable games and their dynamics,, \emph{J. Econom. Theory}, 144 (2009), 1665. doi: 10.1016/j.jet.2009.01.007. Google Scholar

[10]

Z. Hou, Permanence criteria for Kolmogorov systems with delays,, \emph{Proc. Roy. Soc. Edinburgh-A}, 144 (2014), 511. doi: 10.1017/S0308210512000297. Google Scholar

[11]

Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems,, \emph{Dynam. Systems}, 26 (2011), 367. doi: 10.1080/14689367.2011.554384. Google Scholar

[12]

Z. Hou and S. Baigent, Heteroclinic limit cycles in competitive Kolmogorov systems,, \emph{Discrete and Continuous Dynamical Systems}, 33 (2013), 4071. doi: 10.3934/dcds.2013.33.4071. Google Scholar

[13]

S-B Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, \emph{Taiwanese J. Math.}, 9 (2005), 157. Google Scholar

[14]

S-B Hsu and T-W Huang, Global stability for a class of predator-prey systems,, \emph{SIAM J. Appl. Math.}, 55 (1995), 763. doi: 10.1137/S0036139993253201. Google Scholar

[15]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry,, \emph{Ergodic Theory Dynam. Systems}, 15 (1995), 121. doi: 10.1017/S0143385700008270. Google Scholar

[16]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134A (2004), 1177. doi: 10.1017/S0308210500003693. Google Scholar

[17]

P. Laurençot and H. V. Roessel, Nonuniversal self-similarity in a coagulation-annihilation model with constant kernels,, \emph{J. Phys. A}, 43 (2010), 1. doi: 10.1088/1751-8113/43/45/455210. Google Scholar

[18]

B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory,, Vol. 189. Cambridge University Press, (2012). doi: 10.1017/CBO9781139026079. Google Scholar

[19]

X. Liang and J. Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems,, \emph{Nonlinearity}, 16 (2003), 1. doi: 10.1088/0951-7715/16/3/301. Google Scholar

[20]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-3-642-72833-4. Google Scholar

[21]

S. H. Saperstone, Semidynamical Systems in Infinite Dimensional Spaces,, Volume 37 of Applied Mathematical Sciences, (1981). Google Scholar

[22]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific, (1996). doi: 10.1142/9789812830548. Google Scholar

[23]

E. O. Voit and M. A. Savageau, Equivalence between S-Systems and Volterra Systems,, \emph{Math. Biosci.}, 78 (1986), 47. doi: 10.1016/0025-5564(86)90030-1. Google Scholar

[24]

E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems,, \emph{Trans. Amer. Math. Soc.}, 355 (2003), 713. doi: 10.1090/S0002-9947-02-03103-3. Google Scholar

[25]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, \emph{Dynam. Stability Systems}, 8 (1993), 189. doi: 10.1080/02681119308806158. Google Scholar

show all references

References:
[1]

S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems,, \emph{Differential Equations Dynam. Systems}, 20 (2012), 53. doi: 10.1007/s12591-012-0103-0. Google Scholar

[2]

R. M. Goodwin, Chaotic Economic Dynamics,, Clarendon Press, (1990). Google Scholar

[3]

A. S. Hacinliyan, I. Kusbeyzi and O. O. Aybar, Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior,, \emph{Nonlinear Dynam.}, 62 (2010), 17. doi: 10.1007/s11071-010-9695-5. Google Scholar

[4]

R. Haygood, Coexistence in MacArthur-style consumer-resource models,, \emph{Theor. Popul. Biol.}, 61 (2002), 215. Google Scholar

[5]

A. Hastings, Global stability of two species systems,, \emph{J. Math. Biol.}, 5 (1978), 399. doi: 10.1007/BF00276109. Google Scholar

[6]

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

[7]

M. W. Hirsch and H. L. Smith, Monotone dynamical systems,, in \emph{Handbook of Differential Equations: Ordinary Differential Equations 2}, (2006), 1. Google Scholar

[8]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems,, Cambridge University Press, (1998). Google Scholar

[9]

J. Hofbauer, W. H. Sandholm, Stable games and their dynamics,, \emph{J. Econom. Theory}, 144 (2009), 1665. doi: 10.1016/j.jet.2009.01.007. Google Scholar

[10]

Z. Hou, Permanence criteria for Kolmogorov systems with delays,, \emph{Proc. Roy. Soc. Edinburgh-A}, 144 (2014), 511. doi: 10.1017/S0308210512000297. Google Scholar

[11]

Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems,, \emph{Dynam. Systems}, 26 (2011), 367. doi: 10.1080/14689367.2011.554384. Google Scholar

[12]

Z. Hou and S. Baigent, Heteroclinic limit cycles in competitive Kolmogorov systems,, \emph{Discrete and Continuous Dynamical Systems}, 33 (2013), 4071. doi: 10.3934/dcds.2013.33.4071. Google Scholar

[13]

S-B Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, \emph{Taiwanese J. Math.}, 9 (2005), 157. Google Scholar

[14]

S-B Hsu and T-W Huang, Global stability for a class of predator-prey systems,, \emph{SIAM J. Appl. Math.}, 55 (1995), 763. doi: 10.1137/S0036139993253201. Google Scholar

[15]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry,, \emph{Ergodic Theory Dynam. Systems}, 15 (1995), 121. doi: 10.1017/S0143385700008270. Google Scholar

[16]

M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134A (2004), 1177. doi: 10.1017/S0308210500003693. Google Scholar

[17]

P. Laurençot and H. V. Roessel, Nonuniversal self-similarity in a coagulation-annihilation model with constant kernels,, \emph{J. Phys. A}, 43 (2010), 1. doi: 10.1088/1751-8113/43/45/455210. Google Scholar

[18]

B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory,, Vol. 189. Cambridge University Press, (2012). doi: 10.1017/CBO9781139026079. Google Scholar

[19]

X. Liang and J. Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems,, \emph{Nonlinearity}, 16 (2003), 1. doi: 10.1088/0951-7715/16/3/301. Google Scholar

[20]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-3-642-72833-4. Google Scholar

[21]

S. H. Saperstone, Semidynamical Systems in Infinite Dimensional Spaces,, Volume 37 of Applied Mathematical Sciences, (1981). Google Scholar

[22]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific, (1996). doi: 10.1142/9789812830548. Google Scholar

[23]

E. O. Voit and M. A. Savageau, Equivalence between S-Systems and Volterra Systems,, \emph{Math. Biosci.}, 78 (1986), 47. doi: 10.1016/0025-5564(86)90030-1. Google Scholar

[24]

E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems,, \emph{Trans. Amer. Math. Soc.}, 355 (2003), 713. doi: 10.1090/S0002-9947-02-03103-3. Google Scholar

[25]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, \emph{Dynam. Stability Systems}, 8 (1993), 189. doi: 10.1080/02681119308806158. Google Scholar

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