New contractivity condition in a population model with piecewise constant arguments

Abstract

In this paper, we improve contractivity conditions of solutions for the positive equilibrium N ∗ = 1 a + ∑ i = 0 m b i of the following differential equation with piecewise constant arguments: { d N ( t ) d t = N ( t ) r ( t ) { 1 − a N ( t ) − ∑ i = 0 m b i N ( n − i ) } , n ⩽ t < n + 1 , n = 0 , 1 , 2 , … , N ( 0 ) = N 0 > 0 and N ( − j ) = N − j ⩾ 0 , j = 1 , 2 , … , m , where r ( t ) is a nonnegative continuous function on [ 0 , + ∞ ) , r ( t ) ≢ 0 , ∑ i = 0 m b i > 0 , b i ⩾ 0 , i = 0 , 1 , 2 , … , m , and a + b 0 > ∑ i = 1 m b i . 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 ∑ j = 1 m b j − 2 b 0 < a ⩽ ( ∑ j = 1 m b j ) / ( 1 + b 0 / ∑ j = 0 m 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.