# American Institute of Mathematical Sciences

## Phase portraits of the Higgins–Selkov system

 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

Received  August 2020 Revised  November 2020 Published  January 2021

Fund Project: 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)

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.
Citation: Jaume Llibre, Marzieh Mousavi. Phase portraits of the Higgins–Selkov system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021039
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##### References:
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
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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