Global stability of nonautonomous logistic equations with a piecewise constant delay

Abstract

Consider the following nonautonomous differential equation with a piecewise constant delay: { d N ( t ) d t = N ( t ) { r ( t ) − a ( t ) N ( t ) − b ( t ) N ( [ t ] ) } , n ≤ t < n + 1 , n = 0 , 1 , 2 , … , N ( 0 ) = N 0 > 0 , where r ( t ) is a nonnegative continuous function on [ 0 , + ∞ ) , r ( t ) ≢ 0 , a ( t ) ≥ 0 and b ( t ) > 0 . In this paper, using a recent result of a full affirmative answer to the Gopalsamy and Liu’s conjecture for the autonomous case of the above equation with a ( t ) ≡ a ≥ 0 and b ( t ) ≡ b > 0 , we derive three types of sufficient conditions for the global asymptotic stability of solution for the above nonautonomous logistic equation with a piecewise constant delay.