24 research outputs found
Lie symmetries of the Shigesada-Kawasaki-Teramoto system
The Shigesada-Kawasaki-Teramoto system, which consists of two
reaction-diffusion equations with variable cross-diffusion and quadratic
nonlinearities, is considered. The system is the most important case of the
biologically motivated model proposed by Shigesada et al. A complete
description of Lie symmetries for this system is derived. It is proved that the
Shigesada-Kawasaki-Teramoto system admits a wide range of different Lie
symmetries depending on coefficient values. In particular, the Lie symmetry
operators with highly unusual structure are unveiled and applied for finding
exact solutions of the relevant nonlinear system with cross-diffusion
Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results
This review summarizes all known results (up to this date) about methods of
integration of the classical Lotka-Volterra systems with diffusion and presents
a wide range of exact solutions, which are the most important from
applicability point of view. It is the first attempt in this direction. Because
the diffusive Lotka-Volterra systems are used for mathematical modeling
enormous variety of processes in ecology, biology, medicine, physics and
chemistry, the review should be interesting not only for specialists from
Applied Mathematics but also those from other branches of Science. The obtained
exact solutions can also be used as test problems for estimating the accuracy
of approximate analytical and numerical methods for solving relevant boundary
value problems
Lie symmetries of nonlinear parabolic-elliptic systems and their application to a tumour growth model
A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour