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On the impulsive delay hematopoiesis model with periodic coefficients. (English) Zbl 1179.34092
Summary: We consider the nonlinear impulsive delay hematopoiesis model $$p'(t)=\frac{\beta(t)}{1+p^n(t-m\omega)}-\gamma(t)p(t),\quad t\ne t_k,$$ $$p(t^+_k)=(1-b_k)p(t_k),\quad k\in\Bbb N=\{1,2,\dots\},$$ where $n,m\in\Bbb N$, $\beta(t)$, $\gamma(t)$ and $\prod_{0<t_k<t}(1+b_k)$ are positive periodic functions of period $\omega > 0$. We prove that the solutions are bounded and persistent. The persistence implies the survival of the mature cells for a long term. By employing the continuation theorem of coincidence degree, we prove the existence of a positive periodic solution $\overline p(t)$. We establish some sufficient conditions for the global attractivity of $\overline p(t)$. These conditions imply the absence of any disease in the mammal. Moreover, we obtain some sufficient conditions for the oscillation of all positive solutions about the positive periodic solution $\overline p(t)$. These conditions lead to the prevalence of mature cells around the periodic solution. Our results extend and improve some known results in the literature for the autonomous model without impulse. An example is presented to illustrate the main results.

##### MSC:
 34K60 Qualitative investigation and simulation of models 34K11 Oscillation theory of functional-differential equations 34K25 Asymptotic theory of functional-differential equations 34K45 Functional-differential equations with impulses 92C50 Medical applications of mathematical biology
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