Mackey-glass equation with variable coefficients

is considered, with variable coefficients and a nonconstant delay. Under rather natural assumptions all solutions are positive and bounded. Persistence and extinction conditions are presented for this equation. In the case when there exists a constant positive equilibrium, local asymptotic stability of the constant solution and oscillation about this equilibrium are analyzed. The results are illustrated by numerical examples. In particular, it is demonstrated that with delay in both terms, a solution with positive initial conditions may become negative. Q 2006 Elsevier Ltd. All rights reserved. K e y w o r d s D e l a y equations, Extinction and persistence, Mackey-Glass equation, Asymptotics, Oscillation. 1. I N T R O D U C T I O N During the last few decades, it has been recognized that equations with delay more adequately describe various models of mathematical biology than equations without delay. For example, in [1-3], the delay equation (the Mackey-Glass, or the hematopoiesis, equation), dN rNT d~ 1 +-----~ bN, (1) was app l i ed to m o d e l w h i t e b lood cells p r o d u c t i o n . Here, N(t) is t h e d e n s i t y of m a t u r e cells in b lood c i rcu la t ion , t he func t ion , rNr/ (1 + N~) m o d e l e d t h e b lood cell r e p r o d u c t i o n , t h e t ime lag *Partially supported by Israeli Ministry of Absorption. tPartially supported by the NSERC Research Grant and the AIF Research Grant. Author to whom all correspondence should be addressed. 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by .A.h/~-'I~X doff 10.1016/j.camwa.2005.09.001 2 L. ~EREZANSKY AND E. BRAVERMAN NT = N ( t T) described the maturat ional phase before blood cells are released into circulation, the mortali ty rate bN was assumed to be proportional to the circulation. Equation (1) was introduced to explain the oscillations in numbers of neutrophils observed in some cases of chronic myelogenous leukemia [1,2]. The reproduction function can differ from one in (1): for instance, r / ( K ~ + N ~) describes the red blood cells production rate [4], where three parameters r, K, 3 ̀ are chosen to match the experimental data. This leads to the equation d N r d~K ~ + N 7 bN, (2) it describes the feedback function which saturates at low erythrocite numbers and is a decreasing function of increasing red blood cell levels (i.e., negative feedback). Various aspects of autonomous equations (1),(2) and some similar models were studied in [5-11]. The main focus was on the existence of periodic solutions, as well as the existence of apparently aperiodic solutions, iocal and global stability analysis. The summary of some of these results can be found in [12, Sections 4.7-4.9]. Among further developments in this area let us note [13] and recent papers [14-17]. However, in most above-mentioned references autonomous equations were considered (with constant delays and sometimes constant coefficients). In [18], the positiveness of solutions and the global asymptotic stability is studied in some general case, where as an application equation (2) is considered. In the present paper, we s tudy an equation (1) with variable delay and coefficients, d N = r ( t ) N ( g ( t ) ) _ b ( t )N( t ) , (3) dt 1 + [N(g(t))]'r 3' > 0, which in a particular case of constant coefficients r, b and a constant delay g(t) = t T turns into the Mackey-Glass equation. We obtain results on positiveness and boundedness of solutions, on extinction and persistence which extend some results of [12, Sections 4.7-4.9], to equation (1). The paper is organized as follows. In Section 2, after some preliminaries we prove an auxiliary result on linear delay equations with positive and negative coefficients. Section 3 presents main results of the paper: sufficient conditions for positiveness and boundedness of solutions, extinction, and persistence. In Section 4, we s tudy oscillation of solutions about a positive equilibrium (when it exists) and prove the convergence of any nonoscillatory solution to this equilibrium. In Section 5, we illustrate the sharpness of constraints on coefficients by some numerical simulations and discuss the results.