New contractivity condition in a population model with piecewise constant arguments

In this paper, we improve contractivity conditions of solutions for the positive equilibrium N∗=1a+∑i=0mbi of the following differential equation with piecewise constant arguments:{dN(t)dt=N(t)r(t){1−aN(t)−∑i=0mbiN(n−i)},n⩽t0andN(−j)=N−j⩾0,j=1,2,…,m, where r(t)r(t) is a nonnegative continuous function on [0,+∞)[0,+∞), r(t)≢0r(t)≢0, ∑i=0mbi>0, bi⩾0bi⩾0, i=0,1,2,…,mi=0,1,2,…,m, and a+b0>∑i=1mbi. In particular, for the case a=0a=0 and m⩾1m⩾1, we really improve the known three type conditions of the contractivity for solutions of this model (see for example, [Y. Muroya, A sufficient condition on global stability in a logistic equation with piecewise constant arguments, Hokkaido Math. J. 32 (2003) 75–83]). For the other case a≠0a≠0 and m⩾1m⩾1, under the condition ∑j=1mbj−2b0