A predator-prey economic system of tax evasion and corrupt behavior

Figure 1.  Perceived / expected risk of being detected. Source: Special Eurobarometer 402: Undeclared work in the European Union. How would you describe the risk of being detected in the economy https://data.europa.eu/euodp/en/data/dataset/S1080_79_2_402

Figure 2.  Share of firms experiencing at least one bribe payment request (2017 or latest available). Source: World Bank Enterprise Survey. https://doi.org/10.1787/80ebe6e6-en

Figure 3.  The interior equilibrium, $ (\alpha^*, \delta^*) $, into the square $ C = \left\{ (\alpha, \delta): 0 \leq \alpha \leq 1, \; 0 \leq \delta \leq 1\right\} $. The blue line is the nullcline $ \dot{\alpha} = \alpha (0.8+2.5\alpha-2.8\delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(0.4-2\alpha+0.7\delta) = 0 $, the slopes satify $ \frac{b}{c} = \frac{2.5}{2.8}<\frac{e}{f} = \frac{2}{0.7} $. In the figure on the right we can notice that the vertex equilibrium $ (0, 0) $ is a source, but also the upper-border equilibrium $ (\frac{c-a}{b}, 1) = (0.88, 1) $ is unstable, and the other equilibria flee outside the unit square

Figure 4.  The interior equilibrium, $ (\alpha^*, \delta^*) $, into the square $ C = \left\{ (\alpha, \delta): 0 \leq \alpha \leq 1, \; 0 \leq \delta \leq 1\right\} $. The blue line is the nullcline $ \dot{\alpha} = \alpha (1.05+5.5\alpha-4\delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1.05-14.5\alpha+4.5\delta) = 0 $, such that now the basin of attraction towards the socially good state equilibrium $ (1, 0) $ corresponding to taxpaying citizens and non-corrupt auditors, has been enlarged expanded

Figure 5.  The nullclines do not intersect. The blue line is the nullcline $ \dot{\alpha} = \alpha(1.05+4.5 \alpha - 4 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1.05-3.5\alpha+4.5 \delta) = 0 $

Figure 6.  The nullclines do not intersect. The blue line is the nullcline $ \dot{\alpha} = \alpha(1.05+5.5 \alpha - 4 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1.05-7.5\alpha+4.5 \delta) = 0 $

Figure 7.  The nullclines do not intersect, and specifically the nullcline $ \nu $ is outside the unitary square. The blue line is the nullcline $ \dot{\alpha} = \alpha(1.05+2 \alpha - 17 \delta) = 0 $

Figure 8.  The phase portrait on the left shows the transcritical and Saddle-node bifurcation in $ (\alpha^*, \delta^*) = (\alpha_0, \delta_0) = (0, 0) $, the blue line is the nullcline $ \dot{\alpha} = \alpha( \alpha - 2 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(-3\alpha+2 \delta) = 0 $. The phase portrait on the right shows the transcritical and Saddle-node bifurcation in $ (\alpha^*, \delta^*) = (\alpha_1, \delta_1) = (1, 1) $, the blue line is the nullcline $ \dot{\alpha} = \alpha(1+ 1 \alpha - 2 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1-3\alpha+2 \delta) = 0 $

Figure 9.  Saddle-node bifurcation in $ (\alpha^*, \delta^*) = (\alpha_0, \delta_1) = (0, 1) $. The phase portrait on the left shows how the interior equilibrium approaches the vertex equilibrium $ (0, 1) $, where its blue line is the nullcline $ \dot{\alpha} = \alpha(1.4+ 2.7 \alpha - 2 \delta) = 0 $, and its blue dashed line is the nullcline $ \dot{\delta} = \delta(0.8-10 \alpha+1 \delta) = 0 $. While in the two phase portraits on the right the bifurcation conditions are satisfied. There, the blue line is nullcline $ \dot{\alpha} = \alpha(1+ 1 \alpha - 1 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta ( 0.001-8\alpha+0.01\delta) = 0 $