Curve | Region | finite singular point |
$ \alpha_1 $, $ \alpha_2 $ | (1), (2) | stable node |
$ \beta $ ($ b\le b_2 $) | (3) | stable focus |
$ \beta $ ($ b>b_2 $) | (4) | unstable focus |
$ \alpha_3 $ | (5) | unstable node |
$ \gamma $ | empty | |
(6) | saddle |
In this paper we classify the phase portraits in the Poincaré disc of the Selkov model for the glycolysis process
$ \begin{equation*} \dot{x}\, = \, -x+a y+x^2 y, \quad \dot{y} \, = \, b-a y-x^2 y, \end{equation*} $
in function of its parameters $ a, b \in \mathbb{R} $. In particular we determine the regions in the parameter plane with biological meaning, i.e. with $ a $, $ b $, $ x $ and $ y $ positive.
Citation: |
Figure 1. Phase portraits of system (1) with $ a\in \mathbb{R} $ and $ b\ge 0 $ in the Poincaré disc. The shaded areas correspond to the initial conditions of the orbits having a finite final evolution, so these are the initial conditions with biological meaning. In the phase portraits $ E $ and $ I $ the final behaviour is a stable limit cycle, while in the phase portraits $ F $, $ G $ and $ H $ the final behaviour is a singular point
Figure 14. The possible global phase portraits taken from the local phase portrait (Figure 10), for the values of parameters on $ P_3 $, $ P_4 $, $ L_4 $, $ L_6 $, $ L_8 $ and $ R_4 $
Figure 15. The possible global phase portraits taken from the local phase portrait in the region $ R_6 $ (Figure 10)
Table 1. We provide the local phase portrait of the finite singular point when the parameters of the differential system (1) are on the curves or in the regions of Figure 3
Curve | Region | finite singular point |
$ \alpha_1 $, $ \alpha_2 $ | (1), (2) | stable node |
$ \beta $ ($ b\le b_2 $) | (3) | stable focus |
$ \beta $ ($ b>b_2 $) | (4) | unstable focus |
$ \alpha_3 $ | (5) | unstable node |
$ \gamma $ | empty | |
(6) | saddle |
Table 2.
Some approximation of the points on the curves
a | b | on the curve |
$ 0 $ | $ (0.9, 0.91):=b_1 $ | $ \delta_1 $ |
$ -0.001 $ | $ (0.904, 0.905) $ | $ \delta_1 $ |
$ -0.15 $ | $ (1.168, 1.169) $ | $ \delta_1 $ |
$ -1 $ | $ (1.749, 1.75) $ | $ \delta_2 $ |
$ -2 $ | $ (2.154, 2.155) $ | $ \delta_2 $ |
$ -3 $ | $ (2.467, 2.468) $ | $ \delta_2 $ |
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Phase portraits of system (1) with
The regions
Regions and curves separating the different local phase portraits of the finite singular point
Blowing down of the origin of the local chart
Blowing down of the origin of the local chart
Blowing down of the origin of the local chart
Blowing down of the origin of the local chart
The regions
Bifurcation diagram
Local phase portraits at the finite and infinite singular points in the Poincaré disc
The phase portrait of system (4)
The possible global phase portraits taken from the local phase portrait in the region
The direction of vector field for the values of the parameters located in the region
The possible global phase portraits taken from the local phase portrait (Figure 10), for the values of parameters on
The possible global phase portraits taken from the local phase portrait in the region
The direction of vector field for the values of parameters located in the region