Last modified: 2014-03-28
Abstract
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The talk is based on the results published in the recent papers listed below and new unpublished results.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.