Abstract

In this paper we consider the coupled Lotka-Volterra competition-diffusion interaction model. We employ the concept of Lie group theory in constructing the Lie generator, developed the kth order prolongation of the generator of the coupled system. The invariant solution and the new symmetry solutions of the coupled Lotka-Volterra competition-diffusion system are obtained when the diffusive coefficients are equal. The group transformations of solutions of the system are also presented.

References

[1] Mitul I., Bipul I. and Nurul I., Exact solution of the prey-predator model with diffusion using an expansion method. Applied sciences, 1991, vol. 15
[2] Kot M., Elements of mathematical Ecology, Cambridge University press 2001
[3] Kanel J.I. and Zhou L.I., Existence of Wave Front Solutions and Estimates of Wave Speed for A Competition-Diffusion System. Nonlinear Annalysis, Theory, Methods and Applications, 1996, 27
[4] Omar M. H., Application of Symmetry Methods to Partial Differential Equations, PhD thesis 2013
[5] Nikolay A. K. and Anastasia S. Z., Analytical properties and exact solutions of the Lotka-volterra competition system 2014
[6] Murray J.D. Mathematical biology. II. Spatial models and biomedical applications Springer-Verlag, 2003
[7] Li-Chang H., Exact travelling wave solutions for diffusive Lotka-Volterra system of two competing species. Japan, J. indust. Appl. Math. 2012, 29
[8] Kanel J. I.,On the wave front solution of a competition-diffusion system in population dynamics 2006
[9] Yuzo H., Travelling wave for a diffusive Lotka-Volterra competition model I: Singular perturbations, Kyoto Ja , 2003
[10] Ahmed S. and Lazer A. L., An Elementary Approach to Travelling Front Solutions to A System of N Competition-Diffusion Equations, Nonlinear Analysis,Theory, Methods and Applications, 1991, 16.
[11] Wei F., Coexistence, stability, and Limiting Behavior in a One-Predator-Two-Prey Model, Academic Press, Inc, 1993
[12] Kan-On Y., Fisher wave fronts for the Latka-Volterr competition model with diffusive, nonlinear analysis 1997, TMP 28.
[13] Hosono Y, Singular pururbation analysis of travelling waves for Lotka-Volterra competition models, Numerical and Applied Mathematics, Part II, Baltzer, Basel ,1989
[14] Gardner R. A., Existence and stability of travelling wave solution of competition model: a degree theoretic approach, Journal of Differential Equations, 1982, 44.
[15] Morita Y. and Tachibana K., An entire solution to Lotka-Volterra competition-diffusion equation, SIAM Journal on Mathematical Analysis, 2009, 40.
[16] Guo J. S and Wu C. H., Entire solutions for a two-component competition system in Lattice, Tohoku Math. J., 2010, 62
[17] Wang M. and Li G., Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 2010, 23
[18] Rodrigo M. and Mimera M., Exact solutions of competition-diffusion system, Hiroshma Math. J., 2000, 30
[19] Hydon, P., Symmetry methods for differential equations: A beginner’s guide.Cambridge: Cambridge University Press, 2000
[20] Oliveri F., Lie Symmetries of differential equations:Classical results and recent contributions, 2010,Symmetry open access Journals 2, volume 2, Natal, Durban, South Africa
[21] Olver P. J., Applications of Lie Groups to Differential Equations (2nd Ed.) Springer-Verlag, New York, 1993.
[22] Bluman G. W. and Kumei S., Symmetries and Differential Equations, Springer-Verlag, New York 1989.

[23] Ibragimov N., A Practical Course in Mathematical Modelling. ALGA Publications, Blekinge Institute of Technology,Karlskrona,Sweden 2004