On the Impulsive Delay Hematopoiesis Model with Periodic Coefficients

Abstract

In this paper we shall consider the nonlinear impulsive delay hematopoiesis model p0(t) = β(t) 1 + pn(t−mω) − γ(t)p(t), t 6= tk, p(t+k ) = (1 + bk)p(tk), k ∈ N = {1, 2, . . .}, where n,m ∈ N, β(t), γ(t) and 0 0. We prove that the solutions are bounded and persist. The persistence implies the survival of the mature cells for long term. By means of the continuation theorem of coincidence degree, we prove the existence of a positive periodic solution p(t). We also establish some sufficient conditions for the global attractivity of p(t). These conditions imply the absence of any dynamic diseases in the mammal. Moreover, we obtain some sufficient conditions for the oscillation of all positive solutions about the positive periodic solution p(t). These conditions lead to the prevalence of the mature cells around the periodic solution. Our results extend and improve some well known theorems in the literature for the autonomous case without impulse. An example is considered to illustrate the main results.