Qualitative Analysis of a Nonautonomous Nonlinear Delay Differential Equation

This paper is devoted to the systematic study of some qualitative properties of solutions of a nonautonomous nonlinear delay equation, which can be utilized to model single population growths. Various results on the boundedness and oscillatory behavior of solutions are presented. A detailed analysis of the global existence of periodic solutions for the corresponding autonomous nonlinear delay equation is given. Moreover, sufficient conditions are obtained for the solutions to tend to the unique positive equilibrium. Introduction. Using the adsorption theory of chemical kinetics, Ciu and Lawson [1] established the following equation concerning the growth of single populations (i.i) nt Af 1 W f ϊ/γ iΛL -L Λι\Li Λ/γ*. where x(t) is the population density at time t; xm is the maximum value of x allowed by the limiting nutrient, which is equivalent to the so-called carrying capacity; x'm is a parameter which is related to the amount of nutrient and its utilization efficiency by an organism (in units of concentration); μc is a parameter related to the growth velocity (in units of time"), i.e., the so-called intrinsic growth rate. The ratio of xm and x'm is a very important parameter for Equation (1.1). It is assumed that [1-4] 00.5. This AMS Subject Classification 34K15, 92A15.