ABSTRACT
It seems that a happy marriage of the ecology (population dynamics) with the mathematics, using the idea of the global stability of solutions of differential equations, was beginning by V. Volterra. We believe that such idea was in advance of his times and there was no other idea like this. Needless to say, this is not all he has done and it is impossible to get hold of the total of extensive scholary achievements of this super mathematician whom was born in 1860, Ancona of Italy. In this paper, we establish a sufficient condition for the global attractivity of the positive equilibrium of the following delay difference equation
(2) Equation (1) is a discrete analogue of the delay differential equation
(E)
which has been used by Wazewska-Czyzewska and Lasota [7] as a model for the survival of red blood cells in an animal (see also [1,2] and [6]). Here N(t) denotes the density of red
blood cells at time t, 6 is the probability of death of a red blood cell, P and a are positive constants which are related to the production of red blood cells, and r is the time which is required to produce a red blood cell. By the assumption (2), we see that the existence of positive solution {Nn} is guaranteed by No > 0 for equation (1). It is easy to see that equation (1) has a unique equilibrium point TV* > 0, which satisfies the relation by
(3)
We study the global attractivity of the equilibrium point TV* of equation (1). The main result in this paper is the following and this theorem provides a sufficient
condition for the equilibrium point N* of equation (1) to be a global attractor:
Theorem In addition to (2), suppose that
(4) Then, the positive solution {Nn} of equation (1) satisfies
here N* is the one in (3).
