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 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.

[1]

Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923

[2]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[3]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283

[4]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

[5]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[6]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166

[7]

Jun Zhou, Jun Shen. Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3605-3624. doi: 10.3934/dcdsb.2021198

[8]

Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338

[9]

Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291

[10]

Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985

[11]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211

[12]

Zalman Balanov, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree, part I: An axiomatic approach to primary degree. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 983-1016. doi: 10.3934/dcds.2006.15.983

[13]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[14]

Yingjie Bi, Siyu Liu, Yong Li. Periodic solutions of differential-algebraic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1383-1395. doi: 10.3934/dcdsb.2019232

[15]

Xinping Zhou, Yong Li, Xiaomeng Jiang. Periodic solutions in distribution of stochastic lattice differential equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022123

[16]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301

[17]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793

[18]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113

[19]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861

[20]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (2)

[Back to Top]