This talk is devoted to symmetries and exact solutions of the classical Lotka-Volterra systems with diffusion and its highly non-trivial generalisation, the Shigesada–Kawasaki–Teramoto system. Lie and conditional (nonclassical) symmetries are identified and used for constructing exact solutions that satisfy typical boundary conditions and describe different scenario of population (cell, tumour) evolution as time tends to infinity. In particular, several highly nontrivial exact solutions, including traveling fronts, (quasi)periodic solutions and those with separation of variables are found, their properties are identified and a biological interpretation is discussed. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate and numerical methods for solving boundary value prob- lems related to reaction-diffusion systems. Because the Lotka-Volterra type systems are used for mathematical modeling of an enormous variety of processes in ecology, biology, medicine, physics and chemistry, the talk could be interesting not only for specialists in PDEs but also scholars from other branches of science.
The talk is based on the results obtained in collaboration with John R. King (University of Nottingham) and Vasyl’ Davydovych (Institute of Mathematics of NAS of Ukraine) and published in Commun. in Nonlin. Sci. and Num. Simul. 113 (2022) 106579, https://doi.org/10.1016/j.cnsns.2022.106579 and 124 (2023) 107313 https://doi.org/10.1016/j.cnsns.2023.107313