# American Institute of Mathematical Sciences

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

 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

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 $\Gamma$.
Citation: Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar [2] Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree,", AIMS Series on Differential Equations & Dynamical Systems, 1 (2006).   Google Scholar [3] T. Bartsch, "Topological Methods for Variational Problems with Symmetries,", Lecture Notes in Math., 1560 (1993).   Google Scholar [4] N. Hirano and S. Rybicki, Existence of limit cycles for coupled van der Pol equations,, J. Differential Equations, 195 (2003), 194.  doi: 10.1016/S0022-0396(03)00212-2.  Google Scholar [5] N. Hirano and S. Rybicki, Existence of periodic solutions for the Lotka-Volterra type systems,, Nonlinear Aanalysis TMA, (2006).   Google Scholar [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.   Google Scholar [7] K. Gęba, Degree for gradient equivariant maps and equivariant Conley index,, in, 27 (1997), 247.   Google Scholar [8] M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol II,", Applied Mathematical Sciences, 69 (1988).   Google Scholar [9] K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,", Kluwer Academic Publishers, (1992).   Google Scholar [10] J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems,", London Mathematical Society Student Texts, 7 (1988).   Google Scholar [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).   Google Scholar [12] G. E. Hutchinson, "An Introduction to Population Ecology,", Yale University Press, (1978).   Google Scholar [13] J. Ize and A. Vignoli, "Equivariant Degree Theory,", de Gruyter Series in Nonlinear Analysis and Applications, 8 (2003).   Google Scholar [14] R. Levins, "Evolution in Communities Near Equilibrium,", in, (1975).   Google Scholar [15] A. Biglands, Mapl$e^{TM}$ Library Package for the computations of the equivariant degree,, available at \url{http://krawcewicz.net/degree}., ().   Google Scholar

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##### References:
 [1] Z. Balanov, M. Farzamirad and W. Krawcewicz, Symmetric systems of van der Pol equations,, Topol. Methods Nonlinear Anal., 27 (2006), 29.   Google Scholar [2] Z. Balanov, W. Krawcewicz and H. Steinlein, "Applied Equivariant Degree,", AIMS Series on Differential Equations & Dynamical Systems, 1 (2006).   Google Scholar [3] T. Bartsch, "Topological Methods for Variational Problems with Symmetries,", Lecture Notes in Math., 1560 (1993).   Google Scholar [4] N. Hirano and S. Rybicki, Existence of limit cycles for coupled van der Pol equations,, J. Differential Equations, 195 (2003), 194.  doi: 10.1016/S0022-0396(03)00212-2.  Google Scholar [5] N. Hirano and S. Rybicki, Existence of periodic solutions for the Lotka-Volterra type systems,, Nonlinear Aanalysis TMA, (2006).   Google Scholar [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.   Google Scholar [7] K. Gęba, Degree for gradient equivariant maps and equivariant Conley index,, in, 27 (1997), 247.   Google Scholar [8] M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory Vol II,", Applied Mathematical Sciences, 69 (1988).   Google Scholar [9] K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,", Kluwer Academic Publishers, (1992).   Google Scholar [10] J. Hofbauer and K. Sigmund, "The Theory of Evolution and Dynamical Systems,", London Mathematical Society Student Texts, 7 (1988).   Google Scholar [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).   Google Scholar [12] G. E. Hutchinson, "An Introduction to Population Ecology,", Yale University Press, (1978).   Google Scholar [13] J. Ize and A. Vignoli, "Equivariant Degree Theory,", de Gruyter Series in Nonlinear Analysis and Applications, 8 (2003).   Google Scholar [14] R. Levins, "Evolution in Communities Near Equilibrium,", in, (1975).   Google Scholar [15] A. Biglands, Mapl$e^{TM}$ Library Package for the computations of the equivariant degree,, available at \url{http://krawcewicz.net/degree}., ().   Google Scholar
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