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The Shigesada-Kawasaki-Teramoto model: conditional symmetries, exact solutions and their properties

Roman Cherniha, Vasyl' Davydovych, John R. King

Abstract

We study a simplification of the well-known Shigesada-Kawasaki-Teramoto model, which consists of two nonlinear reaction-diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerisation process is suggested and exact solutions are found in this case as well.

The Shigesada-Kawasaki-Teramoto model: conditional symmetries, exact solutions and their properties

Abstract

We study a simplification of the well-known Shigesada-Kawasaki-Teramoto model, which consists of two nonlinear reaction-diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerisation process is suggested and exact solutions are found in this case as well.
Paper Structure (9 sections, 1 theorem, 90 equations, 3 figures, 3 tables)

This paper contains 9 sections, 1 theorem, 90 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Exact solutions and their properties
  4. Case 1 of Table \ref{['tab1']}
  5. Case 9 of Table \ref{['tab1']}
  6. Alternative application of system (\ref{['2-1']})
  7. Conclusion
  8. Acknowledgement
  9. Appendix: Sketch of proof of Theorem \ref{['th3']}

Key Result

Theorem 1

A system of the form (2-1) with the restrictions (2-2) and (2-2*) is invariant under $Q$-conditional symmetry ope-ra-tor(s) (2-29) if and only if this system and the corresponding operator(s) have the forms listed in Table tab1. Any other system of the form (2-1) admitting $Q$-conditional symmetry i with correctly-specified constants $\gamma_i$, $i=1,\dots,6$.

Figures (3)

  • Figure 1: Surfaces representing the $u$ (green) and $v$ (red) components of solution (\ref{['3-10']}) of the SKT system (\ref{['3-0']}) with the parameters $a_1=2, \ a_2=1, \gamma=-2, \ b_2=\frac{1}{10}, \ d_{12}=10c_1-31, C_1=3, \ C_2=2, \ C_3=4, \ x_0=0.$
  • Figure 2: Surfaces representing the $u$ (green) and $v$ (red) components of solution (\ref{['3-10']}) of the SKT system (\ref{['3-0']}) with the parameters $a_1=2, \ a_2=1, \gamma=-\frac{1}{2}, \ b_2=\frac{1}{10}, \ d_{12}=10c_1-16, C_1=3, \ C_2=2, \ C_3=4, \ x_0=0.$
  • Figure 3: Surfaces representing the $u$ (green) and $v$ (red) components of solution (\ref{['3-10']}) of the SKT system (\ref{['3-0']}) with the parameters $a_1=2, \ a_2=1, \gamma=-1, \ b_2=\frac{1}{10}, \ d_{12}=10c_1-21, C_1=3, \ C_2=2, \ C_3=4, \ x_0=0.$

Theorems & Definitions (6)

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