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∗ = 1 a+ m i =0 bi of the following differential equation with piecewise constant arguments: ⎧⎪ ⎨⎪ ⎩ dN(t) dt = N(t)r(t) 1−aN(t) − m i=0 biN(n − i) , n t 0 and N(−j) = N−j 0, j = 1, 2, . . . ,m, where r(t) is a nonnegative continuous function on [0,+∞), r(t) ≡ 0, m i =0 bi > 0, bi 0, i = 0, 1, 2, . . . ,m, and a + b0 > m i =1 bi . In particular, for the case a = 0 and m 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 = 0 and m 1, under the condition mj =1 b j − 2b0 < a ( mj =1 b j)/(1 + b0/ mj =0 b j ), the obtained result partially improves the known results on the contractivity of solutions for the positive equilibrium of this model given by the author [Y. Muroya, Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl. 270 (2002) 602–635] and others