ASYMPTOTIC BEHAVIOR OF A DELAY DIFFERENTIAL MODEL IN POPULATION DYNAMICS

2016. Т. 16, No 2. С. 137–141 137


INTRODUCTION
Consider the following logistic differential equation which is widely used in Population Dynamics = 1 − . Here N(t) is the size of a population, r ≥ 0 is an intrinsic growth rate, K is a carrying capacity or saturation level. A variety of nonlinear differential equations has been developed to construct numerous models of Mathematical Biology [1][2][3].
In order to model processes in nature and engineering it is frequently required to know system states in the past. Depending on the phenomena under study the after-effects represent duration of some hidden processes. In general, DDE's exhibit much more complicated dynamics than ODE's since a time lag can change a stable equilibrium into an unstable one and make populations fluctuate, they provide a richer mathematical framework (compared with ordinary differential equations) for the analysis of biosystems dynamics.
Introduction of complex models of Population Dynamics, based on nonlinear DDE's, has received much attention in the literature in recent years.
The application of delay equations to biomodelling is in many cases associated with studies of dynamic phenomena like oscillations, bifurcations, and chaotic behavior. Time delays represent an additional level of complexity that can be incorporated in a more detailed analysis of a particular system. Delay logistic equation (1) appeared in 1948 in Hutchinson's paper [4]. Here N τ = N(t -τ), τ > 0.
Autonomous equation (1) has been extensively investigated by numerous authors. The first paper on the oscillation of a nonautonomous logistic delay differential equation was published in [5]. Since this publication, the oscillation of the logistic DDE as well as its generalizations were studied by many mathematicians. Some of these results can be found in the monographs [6][7][8].
It is a well-known fact, that the traditional logistic model in some cases produces artificially complex dynamics, therefore it would be reasonable to get away from the specific logistic form in studying population dynamics and use more general classes of growth models.
For example, in order to drop an unnatural symmetry of the logistic curve, we consider the modified logistic form of Pella and Tomlinson [9], [10] or Richards' growth equation with delay (2) According to [9], 0 < γ < 1 for invertebrate populations (examples of invertebrates are insects, worms, starfish, sponges, squid, plankton, crustaceans, and mollusks), and γ ≥ 1 for the vertebrate populations (these include amphibians, birds, fish, mammals, and reptiles).
In [11] the authors considered Eq. (2) with several delays. They obtained conditions for existence of positive solutions and studied so-called long time average stability. In this paper we obtain oscillation and local stability results for nonautonomous Eq. (2) with several delays.

PRELIMINARIES
Our object is a scalar nonlinear delay differential equation under the following conditions: Together with (3) we consider for each t 0 ≥ 0 an initial value problem (5) We also assume that the following hypothesis holds , if it satisfies equation (4) for almost all t ∈ [t 0 , ∞) and equalities (5) for t ≤ t 0 .

OSCILLATION CRITERIA Definition. We say that a function y(t) is nonoscillatory about a number
Eq. (3) has a positive equilibrium * = ∑ . In this section we study oscillation of solutions of (3) about N * . We will present here some lemmas which will be used in this section. Consider the linear delay differential equation and the differential inequalities ( ) + ∑ ( ) ℎ ( ) ≤ 0, ≥ 0, then all the solutions of equation (6) Condition (a3) implies that B k > 0, ∑ = 1 . The zero solution is an equilibrium of Eq. (13), which suits to the equilibrium N * of Eq. (3). By Lemma 1 any solution of (3) is positive. Then for any solution of (13) we have 1 + x(t) > 0. To prove the theorem we have to show that for every nonoscillatory about zero solution of (13) we have lim →∞ ( ) = 0.
(15) Suppose x(t) is a nonoscillatory solution of (13). Without loss of generality we can assume that x(t) > 0, t ≥ 0. Hence (16) If t → +∞ then the right hand side of (16) tends to -∞, the left hand side has a finite limit. This contradiction proves the theorem.
Then all solutions of (3) are oscillatory about N * . Proof. It is sufficient to prove, that all solutions of (13) are oscillatory about zero. Suppose the exists a nonoscillatory solution x of (13). Without loss of generality we can assume, that x(t) > 0, t ≥ 0. Theorem 1 implies, that for some t 0 > 0 and for t ≥ t 0 we have 0 < x(t) < ϵ.
Consider the following function