Global Stability of Discrete Population Models with Time Delays and Fluctuating Environment

Abstract

The global stability of a discrete population model of Volterra type is studied.

Global asymptotic stability beyond 3/2 type stability for a logistic equation with piecewise constant arguments

“…For the case i = 0, a result similar to the global stability is obtained by Muroya [9] and Wang et al [20]. In particular, for the case i = 0, in Muroya [9], the solution has contractivity.…”

“…Then, we obtain one of the main theorems, which extends the results given in [9,20]. then for any solution N(t) of (1.1), lim t→+∞ N(t) = N * and the positive equilibrium N * is globally asymptotically stable.…”

“…In Section 3, using the property of a unimodal function ϕ(x), which is introduced in Section 2, some sufficient conditions for Lemma 2.6 are established. Moreover, we introduce Theorem 3.6, which extends the previous results in [9,20]. Using the previous results given in [19], we finally prove Theorem 1.1 in the last part of Section 3.…”

“…Using the property of the unimodal function ϕ(x), Theorems 3.1-3.3 are derived. Next, we establish one of the main theorems, which extends the results given in [9,20]. Moreover, we introduce useful results from Uesugi et al [19] and finally prove Theorem 1.1.…”

“…In this paper, we consider the following logistic equation with piecewise constant arguments: Several classes of nonlinear delay differential and difference equations have been studied many times in the literature (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references therein) and many authors focus on the global stability of the equation as one of the important topics from the viewpoint of the application. Gopalsamy and Liu [1] studied a class of logistic equations with one piecewise constant argument and offered a conjecture about the global asymptotic stability of the equation.…”