Global attractivity for discrete models of nonautonomous logistic equations

Abstract

Consider the following discrete model of a nonautonomous logistic equation:where c(n) and b j (n), 0 ≤ j ≤ m, n ≥ 0 are bounded andIn this paper, using some kind of iterative method to the above equation, we establish sufficient conditions that ensure the global attractivity for solutions. The result is an extension of the former work [K. Uesugi, Y. Muroya, E. Ishiwata, On the global attractivity for a logistic equation with piecewise constant arguments, J. Math. Anal. Appl. 294 (2004) 560-580] to the nonautonomous case.

The persistence and global attractivity in general nonautonomous discrete single-species Kolmogorov model with delays

“…Assumption ( A 3 ) obviously holds. From Theorem 2.4 we obtain that there exists a constant h > 0 such that for any positive solution x(n) of model (22) there is an integer T 0 such that h x(n) H for all n T , (24) where H = k 1 exp(α 1 (m + ω)) and α 1 = a u .…”

“…Many important results can be found in articles [1][2][3][4][5]7,[9][10][11][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]30,33,34] and the references cited therein. Particularly, the persistence, permanence, global stability and the existence of positive periodic solutions for discrete-time single-species models are studied in articles [2,11,18,22,[25][26][27][28]34].…”

“…2 Next, for model (22), we introduce the following assumption. Let constant H = k 1 exp(α 1 (m + ω)), where…”

“…For model (22), we firstly introduce the following assumption. From assumption (B 1 ), we can choose constant k 1 > 0 and integer T 0 > 0 such that…”

New contractivity condition in a population model with piecewise constant arguments

“…For the case a = 0, r(t) = r (constant) and m 1, Uesugi Muroya and Ishiwata [18] established wide class conditions of global asymptotic stability for the positive equilibrium of this model, and Muroya [13] extend this to the nonautonomous case. Muroya [12] and Muroya, Ishiwata and Guglielmi [14] offer other type of global stability conditions.…”