Persistence and Global Stability in a Population Model

Abstract

A difference equation modelling the dynamics of a population undergoing a density-dependent harvesting is considered. A sufficient condition is established for all positive solutions of the corresponding discrete dynamic system to converge eventually to the positive equilibrium. Elementary methods of differential calculus are used. The result of this article provides a generalization of a result known for a simpler special model with no harvesting.

An affirmative answer to Gopalsamy and Liu's conjecture in a population model

“…(2) if r r * (α), N * is globally asymptotically stable, and they also conjectured that N * is globally asymptotically stable if and only if r r (α) for α ∈ (0, 1) in [4,6].…”

“…(1.1). For reader's convenience, we use the same notations as in [4,11]: It is known in [4] that r * (α) <r(α) for α ∈ (0, 1). For α ∈ (0, 1), Gopalsamy and Liu in [4] showed the following statements:…”

“…The present paper is concerned with the global asymptotic stability of positive equilibrium for the following differential equation ) is a mathematical model to describe a logistically growing population undergoing a densitydependent harvesting and is studied by many authors (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). Clearly, N * = 1 a+b is the unique positive equilibrium of Eq.…”

Permanence, contractivity and global stability in logistic equations with general delays

“…It is assumed in Eq. (1.1) that N(t) denotes the biomass (or population density) of a single species at time t. For m = 0, using elementary methods of differential calculus, Gopalsamy and Liu [3] gave a sufficient condition for all positive solutions of the corresponding discrete dynamic system to converge eventually to the positive equilibrium.…”

Stability switches and Hopf bifurcations of an isolated population model with delay-dependent parameters

“…The studies on the delay differential equations in the population dynamics not only focus on the discussion of stability [2][3][4][5], attractivity [6][7][8] and persistence [9][10][11], but also involve many other dynamical behaviors such as periodic solutions [12][13][14][15][16], bifurcation [17][18][19][20] and chaos [21][22][23][24]. The parameters in most of the aforementioned models are constants which are independent of the time delays.…”