A note on the Lasota discrete model for blood cell production

Figure 2.1.  Graphs of the map $ F $ for $ \gamma = 0.3 $, $ c = 0.6 $, and different values of $ \sigma $: $ \sigma = 0.6<\sigma^*\approx0.774 $ in red ($ F $ is increasing), and $ \sigma = 0.9>\sigma^* $ in blue ($ F $ has two critical points). The dashed line is the graph of $ y = x $

Figure 2.2.  Graphs of the map $ F' $ (solid, blue) and the line $ y = 0 $ (dashed, black), with c = 1. (a): $ \gamma = 0.25<1 $, $ \sigma = 0.8>\sigma^*\approx0.707 $; (b): $ \gamma = 2>1 $, $ \sigma = 0.9>\sigma^*\approx0.841 $. In both cases, there are two intersection points, that determine the critical points of $ F $

Figure 3.1.  Graphs of the map $ F $ for $ \gamma = 8 $, $ \sigma = 0.8 $, and different values of $ c $: $ c = c^*\approx0.425 $ in blue (one positive fixed point), $ c = 0.4<c^* $ in red (no positive fixed points), and $ c = 0.5>c^* $ in black (two positive fixed points). The dashed line is the graph of $ y = x $

Figure 4.1.  Main bifurcation boundaries and regions with different dynamical behavior for equation (1.2) with $ c = 0.47 $, in the parameter plane $ (\gamma,\sigma) $. The two solid lines represent the extinction boundary (red color) and the stability boundary of the largest positive equilibrium (blue color). The vertical dashed line $ \gamma = 1 $ (from $ \sigma = 0 $ to $ \sigma = c = 0.47 $) is the border between global stability of the unique positive equilibrium and a bistability region, in which both the largest positive equilibrium and the extinction equilibrium are asymptotically stable

Figure 4.2.  Bifurcation diagram showing a bubble for equation (1.2) with $ c = 0.47 $, $ \gamma = 7.65 $, using $ \sigma $ as the bifurcation parameter. Black dashed lines correspond to unstable equilibria

Figure 4.3.  Bifurcation diagrams for equation (1.2) with $ c = 0.47 $, using $ \sigma $ as the bifurcation parameter. Black dashed lines correspond to unstable equilibria. For more details, see the text. (a): $ \gamma = 7 $; (b): $ \gamma = 8 $; (c): $ \gamma = 8.5 $ and $ \sigma\in (0,1) $; (d): magnification for $ \gamma = 8.5 $

Figure 4.4.  Bifurcation diagrams for equation (1.2) with $ c = 0.47 $ and different values of $ \sigma $, using $ \gamma $ as the bifurcation parameter. Black dashed lines correspond to unstable equilibria, and red dashed lines to unstable 2-periodic orbits. (a): There are neither oscillations nor extinction windows for $ \sigma = 0.05 $. (b): An extinction window for $ \sigma = 0.1 $. (c): Multiple extinction windows for $ \sigma = 0.9 $