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

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Abstract

In this paper, we consider the following logistic equation with piecewise constant arguments: dN(t)dt=rN(t){1−j=0majN([t−j])},t⩾0,m⩾1,N(0)=N0>0,N(−j)=N−j⩾0,j=1,2,…,m, where r>0, a0,a1,…,am⩾0, ∑j=0maj>0, and [x] means the maximal integer not greater than x. The sequence {Nn}n=0, where Nn=N(n), n=0,1,2,… , satisfies the difference equation Nn+1=Nnexpr1−j=0majNn−j,n=0,1,2,…. Under the condition that the first term a0 dominates the other m coefficients ai, 1⩽im, we establish new sufficient conditions of the global asymptotic stability for the positive equilibrium N=1/(∑j=0maj).

Keywords

Logistic equation
Piecewise constant arguments
Global attractivity

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This work was partially supported by Waseda University Grant for Special Research Projects 2003A-573 and Grant-in-Aid for Young Scientists (B), No. 14740086 of the Ministry of Education, Culture, Sports, Science and Technology of Japan.