Section: New Results

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

In [13], a model for platelet production is introduced for which the platelet count is described by a delay differential equation $P^{\text{'}}\left(t\right)=-\gamma P\left(t\right)+f\left(P\left(t\right)\right)g\left(P(t-r)\right)$ where $f$ and $g$ are positive decreasing functions. First, the authors study the oscillation of the solutions around the unique equilibrium of the equation above, obtaining an inequality implying such an oscillation. They also obtain provide a condition such that this inequality is necessary and sufficient for oscillation. This result is compared to already existing results and the biological meaning of the inequality is studied. The authors also present a result on the asymptotic convergence of the solutions. This result depends on the behavior of the solution for $t\in [0,r]$, and the authors provide an analysis of the link between this behavior and the initial conditions in the case of a simpler model.