Conditional symmetries and exact solutions of diffusive Lotka–Volterra systems

Abstract

Two- and three-component diffusive Lotka–Volterra systems are examined in order to find Q-conditional symmetries, to construct exact solutions and to provide their biological interpretation. An exhaustive description of Q-conditional symmetries of the first type (a special subset of nonclassical symmetries) of these nonlinear systems is derived. An essential part of this chapter is devoted to the construction of exact solutions of the systems in question using the symmetries obtained. Starting from examples of travelling fronts (finding such solutions is important from the applicability point of view), we concentrate mostly on finding exact solutions with a more complicated structure. As a result, a wide range of exact solutions are constructed for the two-component diffusive Lotka–Volterra system and some examples are presented for the three-component diffusive Lotka–Volterra system. Moreover, a realistic interpretation for two and three competing species is provided for some exact solutions.