The talk is devoted to some well-known models based on nonlinear PDEs, in particular, the diffusive Lotka-Volterra (DLV) system and its generalizations. Lie and conditional (nonclassical) symmetries are identified and used for constructing exact solutions, which satisfy typical boundary conditions and describe different scenario of population (cell, tumour ) evolution. In particular, several highly nontrivial exact solutions, including traveling fronts, are found, their properties are identified and a biological interpretation is discussed for the DLV system and the known reaction-diffusion system [K. Aoki et al, Theor. Popul. Biol. 50, 1-17 (1996)] modelling the spread of an initially localized population of farmers into a region occupied by hunter-gatherers. A general scheme for constructing exact solutions of a given nonlinear model is suggested and its applicability is discussed.
The talk is based on new unpublished results and the recent works by R.Cherniha & V.Davydovych, including the monograph “Nonlinear reaction-diffusion systems – conditional symmetry, exact solutions and their applications in biology” (Springer, 2017).