Global stability and repulsion in autonomous Kolmogorov systems

Zhanyuan Hou; Stephen Baigent

doi:10.3934/cpaa.2015.14.1205

Article Contents

2015, Volume 14, Issue 3: 1205-1238. Doi: 10.3934/cpaa.2015.14.1205

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Global stability and repulsion in autonomous Kolmogorov systems

Zhanyuan Hou^1, and
Stephen Baigent^2,

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: August 31, 2014

Revised: November 30, 2014

Published: April 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B25; Secondary: 37C70, 34D23, 34D05.

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References

[1]	S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems, Differential Equations Dynam. Systems, 20 (2012), 53-66.doi: 10.1007/s12591-012-0103-0.
[2]	R. M. Goodwin, Chaotic Economic Dynamics, Clarendon Press, Oxford, 1990.
[3]	A. S. Hacinliyan, I. Kusbeyzi and O. O. Aybar, Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior, Nonlinear Dynam., 62 (2010), 17-26.doi: 10.1007/s11071-010-9695-5.
[4]	R. Haygood, Coexistence in MacArthur-style consumer-resource models, Theor. Popul. Biol., 61 (2002), 215-223.
[5]	A. Hastings, Global stability of two species systems, J. Math. Biol., 5 (1978), 399-403.doi: 10.1007/BF00276109.
[6]	M. W. Hirsch, Systems of differential equations that are competitive or cooperative III: competing species, Nonlinearity, 1 (1988), 51-71.
[7]	M. W. Hirsch and H. L. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations 2, 2006, 1-136.
[8]	J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, New York, 1998.
[9]	J. Hofbauer, W. H. Sandholm, Stable games and their dynamics, J. Econom. Theory, 144 (2009), 1665-1693.doi: 10.1016/j.jet.2009.01.007.
[10]	Z. Hou, Permanence criteria for Kolmogorov systems with delays, Proc. Roy. Soc. Edinburgh-A, 144 (2014), 511-531.doi: 10.1017/S0308210512000297.
[11]	Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems, Dynam. Systems, 26 (2011), 367-390.doi: 10.1080/14689367.2011.554384.
[12]	Z. Hou and S. Baigent, Heteroclinic limit cycles in competitive Kolmogorov systems, Discrete and Continuous Dynamical Systems, 33 (2013), 4071-4093.doi: 10.3934/dcds.2013.33.4071.
[13]	S-B Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. Math., 9 (2005), 157-173.
[14]	S-B Hsu and T-W Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.doi: 10.1137/S0036139993253201.
[15]	M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynam. Systems, 15 (1995), 121-147.doi: 10.1017/S0143385700008270.
[16]	M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II, Proc. Roy. Soc. Edinburgh Sect. A, 134A (2004), 1177-1197.doi: 10.1017/S0308210500003693.
[17]	P. Laurençot and H. V. Roessel, Nonuniversal self-similarity in a coagulation-annihilation model with constant kernels, J. Phys. A, 43 (2010), 1-10.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, Nonlinearity, 16 (2003), 1-18.doi: 10.1088/0951-7715/16/3/301.
[20]	K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin, 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, Springer-Verlag Gmbh, 1981.
[22]	Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, Singapore, 1996.doi: 10.1142/9789812830548.
[23]	E. O. Voit and M. A. Savageau, Equivalence between S-Systems and Volterra Systems, Math. Biosci., 78 (1986), 47-55.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, Trans. Amer. Math. Soc., 355 (2003), 713-734.doi: 10.1090/S0002-9947-02-03103-3.
[25]	M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217.doi: 10.1080/02681119308806158.