Existence of nonstationary periodic solutions for \Gamma-symmetric Lotka-Volterra type systems

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August 2011, 30(3): 709-735. doi: 10.3934/dcds.2011.30.709

Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems

Norimichi Hirano ^1, , Wieslaw Krawcewicz ^2, and Haibo Ruan ^3,

1.	Department of Mathematics, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama, Japan
2.	Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, United States
3.	Department of Mathematics, Universität Hamburg, 20146 Hamburg, Germany

Received February 2010 Revised October 2010 Published March 2011

Abstract
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In this paper we present a general framework for applications of the twisted equivariant degree (with one free parameter) to an autonomous $\Gamma$-symmetric system of functional differential equations in order to detect and classify (according to their symmetric properties) its periodic solutions. As an example we establish the existence of multiple non-constant periodic solutions of delay Lotka-Volterra equations with $\Gamma$-symmetries. We also include some computational examples for several finite groups $\Gamma$.

Keywords: periodic solutions, equivariant degree., Autonomous functional differential equations.

Mathematics Subject Classification: Primary: 34C25, 37L20, 47H11; Secondary: 34L30, 37N2.

Citation: Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 709-735. doi: 10.3934/dcds.2011.30.709

References:

[1]	Z. Balanov, M. Farzamirad and W. Krawcewicz, Symmetric systems of van der Pol equations, Topol. Methods Nonlinear Anal., 27 (2006), 29-90.
[2]	Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree," AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006.
[3]	T. Bartsch, "Topological Methods for Variational Problems with Symmetries," Lecture Notes in Math., 1560, Springer-Verlag, Berlin, 1993.
[4]	N. Hirano and S. Rybicki, Existence of limit cycles for coupled van der Pol equations, J. Differential Equations, 195 (2003), 194-209. doi: 10.1016/S0022-0396(03)00212-2.
[5]	N. Hirano and S. Rybicki, Existence of periodic solutions for the Lotka-Volterra type systems, Nonlinear Aanalysis TMA, (2006).
[6]	G. Dylawerski, K. Gęba, J. Jodel and W. Marzantowicz, An $S^1$-equivariant degree and the Fuller index, Ann. Polon. Math., 52 (1991), 243-280.
[7]	K. Gęba, Degree for gradient equivariant maps and equivariant Conley index, in "Topological Nonlinear Analysis II" (Frascati, 1995), Progr. Nonlinear Differential Equations Appl., 27, Birkh\"auser Boston, Boston, (1997), 247-272.
[8]	M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol II," Applied Mathematical Sciences, 69, Springer-Verlag, New York-Berlin, 1988.
[9]	K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Kluwer Academic Publishers, 1992.
[10]	J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," London Mathematical Society Student Texts, 7, Cambridge University Press, Cambridge, 1988.
[11]	K. P. Hadeler and G. Bocharov, Where to put delays in population models, in particular in the neutral case, Canadian Applied Mathematics Quarterly, 11 (2003).
[12]	G. E. Hutchinson, "An Introduction to Population Ecology," Yale University Press, New Haven, 1978.
[13]	J. Ize and A. Vignoli, "Equivariant Degree Theory," de Gruyter Series in Nonlinear Analysis and Applications, 8, Walter de Gruyter & Co., Berlin-New York, 2003.
[14]	R. Levins, "Evolution in Communities Near Equilibrium," in "Ecology and Evolution of Communities" (M. L. Cody and J. M. Diamond, eds.), Harvard University Press, 1975.
[15]	A. Biglands, Mapl$e^{TM}$ Library Package for the computations of the equivariant degree, available at http://krawcewicz.net/degree.

show all references

References:

[1]	Z. Balanov, M. Farzamirad and W. Krawcewicz, Symmetric systems of van der Pol equations, Topol. Methods Nonlinear Anal., 27 (2006), 29-90.
[2]	Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree," AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006.
[3]	T. Bartsch, "Topological Methods for Variational Problems with Symmetries," Lecture Notes in Math., 1560, Springer-Verlag, Berlin, 1993.
[4]	N. Hirano and S. Rybicki, Existence of limit cycles for coupled van der Pol equations, J. Differential Equations, 195 (2003), 194-209. doi: 10.1016/S0022-0396(03)00212-2.
[5]	N. Hirano and S. Rybicki, Existence of periodic solutions for the Lotka-Volterra type systems, Nonlinear Aanalysis TMA, (2006).
[6]	G. Dylawerski, K. Gęba, J. Jodel and W. Marzantowicz, An $S^1$-equivariant degree and the Fuller index, Ann. Polon. Math., 52 (1991), 243-280.
[7]	K. Gęba, Degree for gradient equivariant maps and equivariant Conley index, in "Topological Nonlinear Analysis II" (Frascati, 1995), Progr. Nonlinear Differential Equations Appl., 27, Birkh\"auser Boston, Boston, (1997), 247-272.
[8]	M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol II," Applied Mathematical Sciences, 69, Springer-Verlag, New York-Berlin, 1988.
[9]	K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Kluwer Academic Publishers, 1992.
[10]	J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems," London Mathematical Society Student Texts, 7, Cambridge University Press, Cambridge, 1988.
[11]	K. P. Hadeler and G. Bocharov, Where to put delays in population models, in particular in the neutral case, Canadian Applied Mathematics Quarterly, 11 (2003).
[12]	G. E. Hutchinson, "An Introduction to Population Ecology," Yale University Press, New Haven, 1978.
[13]	J. Ize and A. Vignoli, "Equivariant Degree Theory," de Gruyter Series in Nonlinear Analysis and Applications, 8, Walter de Gruyter & Co., Berlin-New York, 2003.
[14]	R. Levins, "Evolution in Communities Near Equilibrium," in "Ecology and Evolution of Communities" (M. L. Cody and J. M. Diamond, eds.), Harvard University Press, 1975.
[15]	A. Biglands, Mapl$e^{TM}$ Library Package for the computations of the equivariant degree, available at http://krawcewicz.net/degree.

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