On Gopalsamy and Liu's conjecture for global stability in a population model

摘要

For a logistic equation with a piecewise constant argument which models the dynamics of a population of a single species undergoing a density-dependent harvesting, Gopalsamy and Liu (J. Math. Anal. Appl. 224 (1998) 59–80), have offered a condition of the growth rate of species and conjectured that this is a necessary and sufficient condition of not only the asymptotic stability but also the global asymptotic stability for the positive equilibrium of the equation. But until now, there were no mathematical answers except computer simulations.In this paper, we establish a mathematically rigorous proof of a partial affirmative answer of this conjecture.