Global stability and chaos in a population model with piecewise constant arguments

On the global attractivity for a logistic equation with piecewise constant arguments

“…By Gopalsamy [4], for (2.3)-(2.5) with r 2 = 0, (3.2) implies lim n→∞ x n = 0. 2 Lemma 3.1 implies that if 0 < r = r 1 2 and r 2 = 0, then lim n→∞ x n = 0.…”

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

“…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.…”

“…(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].…”

Oscillation and stability of nonlinear discrete models exhibiting the Allee effect

“…One of the ways of deriving difference equations modeling the dynamic of populations with nonoverlapping generations is based on appropriate modifications of models with overlapping generations. In this approach, differential equations with piecewise constant arguments have been useful, see for example the paper by L i u and G o p a l s a m y [13]. Recently the method that has been established by L i u and G o p a l s a m y has been used by some authors to find the discrete analogy of some mathematical models.…”

“…Recently the method that has been established by L i u and G o p a l s a m y has been used by some authors to find the discrete analogy of some mathematical models. Using the technique that has been used in [13], we will derive the discrete analogy of equation (3). Thinking of differential equations with piecewise constant arguments, we can go on with the discrete analogy of equation (3).…”