On the global attractivity for a logistic equation with piecewise constant arguments

Abstract

In this paper, we consider the following logistic equation with piecewise constant arguments: dN(t) dt =rN(t){1− ∑ j=0 m a jN([t−j])}, t⩾0, m⩾1, N(0)=N 0>0, N(−j)=N −j⩾0, j=1,2,…,m, where r>0, a 0, a 1,…, a m ⩾0, ∑ j=0 m a j >0, and [ x] means the maximal integer not greater than x. The sequence { N n } n=0 ∞, where N n = N( n), n=0,1,2,… , satisfies the difference equation N n+1=N n exp r 1− ∑ j=0 m a jN n−j , n=0,1,2,…. Under the condition that the first term a 0 dominates the other m coefficients a i , 1⩽ i⩽ m, we establish new sufficient conditions of the global asymptotic stability for the positive equilibrium N ∗=1/(∑ j=0 ma j) .