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Symmetries and exact solutions of the diffusive Holling-Tanner prey-predator model

Roman Cherniha, Vasyl' Davydovych

Abstract

We consider the classical Holling-Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling-Tanner system is studied by means of symmetry based methods. Lie and Q-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.

Symmetries and exact solutions of the diffusive Holling-Tanner prey-predator model

Abstract

We consider the classical Holling-Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling-Tanner system is studied by means of symmetry based methods. Lie and Q-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.
Paper Structure (6 sections, 3 theorems, 97 equations, 6 figures, 2 tables)

This paper contains 6 sections, 3 theorems, 97 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lie symmetry classification and reduction to ODEs
  3. Lie solutions of the DHT model
  4. Conditional symmetries of the DHT model (\ref{['1-1']})
  5. Non-Lie exact solutions of the DHT model
  6. Conclusions

Key Result

Theorem 1

The DHT system (1-1) with arbitrary parameters $A, \ R, \ S$ and $d$ is invariant with respect to two-dimensional Lie algebra of invariance (2-1) . This system admits three-dimensional or higher-dimensional Lie algebras of invariance if and only if its parameters have the forms listed in Table tab1.

Figures (6)

  • Figure 1: Surfaces representing the $u$ (green) and $v$ (red) components of solution (\ref{['2-34*']}) of the DHT system (\ref{['2-11']}) with the parameters $d=1, \ S=3, \ R=1.5, \ t_0=0.1.$
  • Figure 2: Surfaces representing the $u$ (green) and $v$ (red) components of solution (\ref{['2-34*']}) of the DHT system (\ref{['2-11']}) with the parameters $d=1, \ S=3, \ R=0.5, \ t_0=0.1.$
  • Figure 3: Surfaces representing the $u$ (green) and $v$ (red) components of the exact solution (\ref{['4-16']}) of the DHT system (\ref{['3-8']}). The parameters $S=2, \ C=-0.25$.
  • Figure 4: Surfaces representing the $u$ (green) and $v$ (red) components of the exact solution (\ref{['4-16']}) of the DHT system (\ref{['3-8']}). The parameters $S=1.4, \ C=-0.125$.
  • Figure 5: Surface representing the component $U$ of the approximate solution (\ref{['4-20']}) of the DHT systems (\ref{['3-8']}). The parameters $S=2, \ C=-0.35$, $x_0=-30, \ x_1=0, \ x_2=30, \ t_0=-1, \ t_1=0, t_2=1.$
  • ...and 1 more figures

Theorems & Definitions (9)

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