59 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
Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations
Q-conditional symmetries (nonclassical symmetries) for a general class of
two-component reaction-diffusion systems with constant diffusivities are
studied. Using the recently introduced notion of Q-conditional symmetries of
the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207),
an exhaustive list of reaction-diffusion systems admitting such symmetry is
derived. The form-preserving transformations for this class of systems are
constructed and it is shown that this list contains only non-equivalent
systems. The obtained symmetries permit to reduce the reaction-diffusion
systems under study to two-dimensional systems of ordinary differential
equations and to find exact solutions. As a non-trivial example, multiparameter
families of exact solutions are explicitly constructed for two nonlinear
reaction-diffusion systems. A possible interpretation to a biologically
motivated model is presented
A hunter-gatherer-farmer population model: Lie symmetries, exact solutions and their interpretation
The Lie symmetry classification of the known three-component
reaction-diffusion system modelling the spread of an initially localized
population of farmers into a region occupied by hunter-gatherers is derived.
The Lie symmetries obtained for reducing the system in question to systems of
ODEs and constructing exact solutions are applied. Several exact solutions of
traveling front type are found, their properties are identified and biological
interpretation is discussed
Multi-parameter reaction–diffusion systems with quadratic nonlinearity and delays: new exact solutions in elementary functions
The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays
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