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Global dynamics and integrability of a Leslie-Gower predator-prey model with linear functional response and generalist predator

Martha Alvarez-Ramírez, Johanna D. García-Saldaña, Mario Medina

Abstract

We deal with a Leslie-Gower predator-prey model with a generalist or alternating food for predator and linear functional response. Using a topological equivalent polynomial system we prove that the system is not Liouvillian (hence also not Darboux) integrable. In order to study the global dynamics of this model, we use the Poincaré compactification of $\mathbb{R}^2$ to characterize all phase portraits in the Poincaré disc, obtaining two different topological phase portraits.

Global dynamics and integrability of a Leslie-Gower predator-prey model with linear functional response and generalist predator

Abstract

We deal with a Leslie-Gower predator-prey model with a generalist or alternating food for predator and linear functional response. Using a topological equivalent polynomial system we prove that the system is not Liouvillian (hence also not Darboux) integrable. In order to study the global dynamics of this model, we use the Poincaré compactification of to characterize all phase portraits in the Poincaré disc, obtaining two different topological phase portraits.

Paper Structure

This paper contains 9 sections, 12 theorems, 32 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and statement of results
  2. Proof of the Theorem \ref{['Theo1']}
  3. Proof of the Theorem \ref{['Theo2']}
  4. The finite equilibrium points
  5. The infinite equilibrium points
  6. Global phase portrait of system \ref{['eq4']}
  7. Concluding Remarks
  8. Invariant algebraic curves and Liouvillian integrability
  9. Poincaré compactification in $\mathbb{R}^2$

Key Result

Theorem 1

System eq4 is not Liouvillian integrable.

Figures (4)

  • Figure 1: Global phase portrait of system \ref{['eq4']} at infinity, shown in the closed positive quadrant of the Poincaré disc.
  • Figure 2: Topological local phase portrait of system \ref{['Poincare2']}.
  • Figure 3: The $u$-directional blow-up of the local chart $U_2$ of system \ref{['blowup-u']}.
  • Figure 4: The $v$-directional blow-up of the local chart $U_2$ of system \ref{['blowup-u']}.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 7 more