Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Symmetry based methods for constructing exact solutions of the diffusive Lotka-Volterra system

Last modified: 2014-03-28


The reaction-diffusion systems of the Lotka-Volterra type is the most common systems for modeling different types of interaction between species. Nevertheless the classical Lotka--Volterra  system (without diffusion terms) was introduced about  90 years ago, its different generalizations are  widely studied at the present time because oftheir importance for  mathematical modeling various   processes in population dynamics and  ecology.  In this talk, some recent results for two- and three-component diffusive Lotka-Volterra systems, which were derived by symmetry based methods,  are presented. At the first step, Lie  and   conditional symmetries  in the form of linear first-order differential operators for the systems in question are constructed. The next step consists in application of the symmetries obtained in order to reduce     the  diffusiveLotka-Volterra systems (with correctly-specified coefficients)  to the systems of ordinary differential equations (ODE). Solving the ODE systems obtained, a wide range of exact solutions for the diffusive Lotka-Volterra systems are found. Finally, an analysis of some  exact solutions  are presented, in particular, this is shown that they    describe   different scenarios  of competition between  populations in the case  of two or three  species.

The talk is based on the results  published in the recent papers listed below and new unpublished results.

1. Cherniha R and  Davydovych V. Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system. Math Comput Modelling (2011); vol.54, P.1238-51.  2. Cherniha R. and Davydovych V. Lie and conditional symmetries of the three-component diffusive Lotka-Volterra system. J  Phys A: Math and Theor (2013); vol.46, 185204  (18pp).

Acknowledgment: This work was  supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme.


Lotka-Volterra system; exact solution; conditional symmetry