Oscillations and asymptotic convergence for a delay differential equation modeling platelet production

Figure 1.  Model of Megakaryopoiesis. The linear differentiation process, starting from HSC and ending with platelets, is positively regulated by TPO. The quantity of TPO is in turn modulated by the number of platelets: the more platelets, the less circulating TPO

Figure 2.  Oscillations appear when $\alpha_A$ increases. As $\alpha_A$ (the maximum number of platelets that a megakaryocyte can shed, see Equation (6)) increases, $R = rqe^{r\left( \gamma +p\right) }-\frac{1}{e}$ becomes positive and $x$ (blue) starts to oscillate around $x^*$ (dashed red). Black marks are placed where $x(t)$ goes through $x^*$. (A) $\alpha_A = 5000, R = -0.0492$ and there are no oscillations. (C) $\alpha_A = 10000, R = 7.6863$ and there are oscillations. (B) $\alpha_A = 20000, R = 83$ and there are oscillations

Figure 3.  Solutions of (18) with or without low initial slope. (Top) The solution goes through $x(t) = x^*$ after $t = r$, it meets the low initial slope criterion. (Bottom) The solution goes through $x(t) = x^*$ before $t = r$, it does not meet the low initial slope criterion

Figure 4.  An example of sequences $(y_n)_{n\in \mathbb{N}}$ and $(z_n)_{n\in \mathbb{N}}$. The decreasing (resp. increasing) sequence $(z_n)_{n\in \mathbb{N}}$ (resp. $(y_n)_{n\in \mathbb{N}}$) bounds $x(t)$ for $t>t_{2n}^*$ (resp. for $t>t_{2n-1}$)

Figure 5.  $ {\bf{Simplified \ model \ of \ Megakaryopoiesis}}$

Figure 6.  Initial slope and initial conditions. Four solutions of the equation (44) (blue) where different initial conditions lead to different relative position for $\tau$, $r$ (dashed green) and the time $t_0$ when $P(t)$ crosses $P^*$ (dashed red).