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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

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.

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

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