Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symmetries and exact solutions of a reaction-diffusion system arising in population dynamics

Philip Broadbridge, Roman Cherniha, Vasyl' Davydovych, Ian Marquette

TL;DR

A general algorithm for finding Q-conditional symmetries of nonlinear evolution systems of the most general form is presented in a useful form for other researchers.

Abstract

A system of two cubic reaction-diffusion equations for two independent gene frequencies arising in population dynamics is studied. Depending on values of coefficients, all possible Lie and $Q$-conditional (nonclassical) symmetries are identified. A wide range of new exact solutions is constructed, including those expressible in terms of a Lambert function and not obtainable by Lie symmetries. An example of a new real-world application of the system is discussed. A general algorithm for finding Q-conditional symmetries of nonlinear evolution systems of the most general form is presented in a useful form for other researchers.

Symmetries and exact solutions of a reaction-diffusion system arising in population dynamics

TL;DR

A general algorithm for finding Q-conditional symmetries of nonlinear evolution systems of the most general form is presented in a useful form for other researchers.

Abstract

A system of two cubic reaction-diffusion equations for two independent gene frequencies arising in population dynamics is studied. Depending on values of coefficients, all possible Lie and -conditional (nonclassical) symmetries are identified. A wide range of new exact solutions is constructed, including those expressible in terms of a Lambert function and not obtainable by Lie symmetries. An example of a new real-world application of the system is discussed. A general algorithm for finding Q-conditional symmetries of nonlinear evolution systems of the most general form is presented in a useful form for other researchers.

Paper Structure

This paper contains 12 sections, 1 theorem, 108 equations.

Table of Contents

  1. Introduction
  2. Lie and conditional symmetries: main results
  3. Exact solutions of the RD system (\ref{['2-2**']})
  4. Lie's solutions
  5. Non-Lie solutions
  6. General algorithm for finding $Q$-conditional symmetries of evolution systems
  7. Solving the determining equations for the RD system (\ref{['2-2']})
  8. Case (3Ia)
  9. Case (3IIa)
  10. An example: settling in a new mining town
  11. Conclusion
  12. Appendix A: Determining systems of equation generic form

Key Result

Theorem 1

The system RD (2-2) admits a $Q$-conditional symmetry of the form (2-3) with $\xi^0\not=0$ only in the case $d_1\not=d_2, \ B_1=B_2=0$ and $A_1d_2=A_2d_1$, i.e. The most general form of the $Q$-conditional symmetry ($\xi^0\not=0$) of system (2-4) is given by the formula where $A$, $\alpha$, $\beta$, $\gamma$ are arbitrary constants and $\sigma = \gamma \frac{d_2 -d_1}{2 d_1 d_2}.$

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Remark 4