Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global phase portraits of a predator-prey system

Érika Diz-Pita, Jaume Llibre, M. V. Otero-Espinar

TL;DR

This work delivers a comprehensive classification of the global dynamics for a family of Kolmogorov predator–prey systems with Holling type II interactions by applying Poincaré compactification to the positive quadrant. The authors analyze finite and infinite singularities, establish Hopf bifurcations at the interior equilibrium $P_2$, and use Bendixson–Dulac arguments to delineate parameter regions with and without limit cycles, resulting in a trichotomy of global portraits (A,B,C) across regions bounded by surfaces $S_1,S_2,S_3$. The key contributions include a complete catalog of possible global phase portraits, identification of a unique stable limit cycle in Hopf-bifurcation regimes, and precise conditions under which cycles exist or are precluded, with a conjecture addressing unresolved subregions. These findings enhance understanding of predator–prey dynamics under Kolmogorov polynomial representations and provide a rigorous framework for predicting long-term ecological behavior in the positive quadrant.

Abstract

We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator-prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems.

Global phase portraits of a predator-prey system

TL;DR

This work delivers a comprehensive classification of the global dynamics for a family of Kolmogorov predator–prey systems with Holling type II interactions by applying Poincaré compactification to the positive quadrant. The authors analyze finite and infinite singularities, establish Hopf bifurcations at the interior equilibrium , and use Bendixson–Dulac arguments to delineate parameter regions with and without limit cycles, resulting in a trichotomy of global portraits (A,B,C) across regions bounded by surfaces . The key contributions include a complete catalog of possible global phase portraits, identification of a unique stable limit cycle in Hopf-bifurcation regimes, and precise conditions under which cycles exist or are precluded, with a conjecture addressing unresolved subregions. These findings enhance understanding of predator–prey dynamics under Kolmogorov polynomial representations and provide a rigorous framework for predicting long-term ecological behavior in the positive quadrant.

Abstract

We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator-prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems.
Paper Structure (8 sections, 5 theorems, 37 equations, 3 figures, 1 table)

This paper contains 8 sections, 5 theorems, 37 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite Singular Points
  4. Infinite Singular Points
  5. Cases with no singular points in the positive quadrant
  6. Cases with singular points in the positive quadrant
  7. Existence of limit cycles
  8. Phase portraits on the positive quadrant of the Poincaré disc

Key Result

Theorem 1.1

The global phase portait of system sis in the closed positive quadrant of the Poincaré disc is topologically equivalent to one of the $3$ phase portraits of Figure fig:globales in the following way:

Figures (3)

  • Figure 1: Phase portraits of system \ref{['sis']} in the positive quadrant of the Poincaré disc.
  • Figure 2: The regions I, II-a, II-b, III and the surfaces separating the different phase portraits: $S_1:\left\lbrace \delta=c/(b+1) \mid b, c \geq 0 \right\rbrace$, $S_2:\left\lbrace \delta= (1+c-b)/(b+1) \mid b,c \geq 0, (1+c-b)/(b+1)< c/(b+1) \right\rbrace$ and $S_3:\left\lbrace \delta= c(1-b)/(1+b) \mid b,c \geq 0 \right\rbrace$
  • Figure 3: Desingularization of the origin of system \ref{['sisU2']}.

Theorems & Definitions (12)

  • Theorem 1.1
  • Conjecture
  • Theorem 6.1
  • proof
  • Theorem 6.2
  • proof
  • Proposition 6.3
  • proof
  • Remark 6.4
  • Theorem 6.5
  • ...and 2 more