An affirmative answer to the extended Gopalsamy and Liu's conjecture on the global asymptotic stability in a population model

作者：Li, Huaixing^*; Yuan, Rong

来源：Nonlinear Analysis: Real World Applications , 2010, 11(5): 3295-3308.

DOI：10.1016/j.nonrwa.2009.11.022

• Global asymptotic stability
• Logistic equation
• Conjecture

We study the global asymptotic stability of the positive equilibrium in a population model with a piecewise constant argument. Gopalsamy and Liu conjectured that the positive equilibrium N* = 1/a+b is globally asymptotically stable if and only if the following inequality holds,
r <= (r) over bar(r) over cap(alpha) 1+alpha/alpha ln 1+alpha/1-alpha
which has been solved by Muroya and Kato (2005)[2], Li and Yuan (2008)[1] for alpha := a/b is an element of [0.1). But, for alpha is an element of(-1, 0), is the above inequality the necessary and sufficient condition for the global asymptotic stability of the positive equilibrium 1/a+b ? In this paper, we will give an affirmative answer to the extended Gopalsamy and Liu's conjecture.

• 出版日期2010-10
• 单位北京师范大学