Phase portraits of the Higgins–Selkov system

Jaume Llibre; Marzieh Mousavi

doi:10.3934/dcdsb.2021039

Article Contents

2022, Volume 27, Issue 1: 245-256. Doi: 10.3934/dcdsb.2021039

This issue Previous Article The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays Next Article Random perturbations of an eco-epidemiological model

Phase portraits of the Higgins–Selkov system

Jaume Llibre^1, and
Marzieh Mousavi^2, ,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2.
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

^* Corresponding author: Marzieh Mousavi
^* Corresponding author: Marzieh Mousavi

Received: July 31, 2020

Revised: October 31, 2020

Early access: January 2021

Published: January 2022

The first author is partially supported by the Ministerio de Economìa, Industria y competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d'Ajusts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics-H2020-MSCA-RISE-2017-777911. The second author is supported by Isfahan University of Technology (IUT)

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Abstract

In this paper we study the dynamics of the Higgins–Selkov system

$ \begin{equation*} \dot{x} = 1-xy^\gamma, \quad\dot{y} = \alpha y(xy^{\gamma -1}-1), \end{equation*} $

where $ \alpha $ is a real parameter and $ \gamma>1 $ is an integer. We classify the phase portraits of this system for $ \gamma = 3, 4, 5, 6, $ in the Poincaré disc for all the values of the parameter $ \alpha $. Moreover, we determine in function of the parameter $ \alpha $ the regions of the phase space with biological meaning.

Keywords:

Mathematics Subject Classification: Primary: 34C05, 34C07, 34C08.

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Figure 1. The phase portraits of system (1) for $ \gamma = 3\; \text{and}\; 5 $ in the Poincaré disc. The shaded areas correspond to the initial conditions of the orbits having a final finite evolution, so these are the initial conditions with biological meaning. In the phase portrait (c) the final behaviour is a stable singular point, and in the phase portrait (d) is a stable limit cycle

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Figure 2. The phase portraits of system (1) for $ \gamma = 4\; \text{and}\; 6 $ in the Poincaré disc. The shaded areas correspond to the initial conditions of the orbits having a final finite evolution, so these are the initial conditions with biological meaning. In the phase portrait (c) the final behaviour is a stable singular point, and in the phase portrait (d) is a stable limit cycle

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