Boundedness of positive solutions of a system of nonlinear delay differential equations

Abstract

In this manuscript the system of nonlinear delay differential equationsẋ i (t) = n j=1 n 0 =1 α ij (t)h ij (x j (t − τ ij (t))) − β i (t)f i (x i (t)) + ρ i (t), t ≥ 0, 1 ≤ i ≤ n is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the system. In several particular cases, explicit formulas are given for the estimates of the upper and lower limit of the solutions. In a special case, the asymptotic equivalence of the solutions is investigated.

Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms

“…We also emphasize that typically the nonlinearites f i (t, φ) in (1) are not monotone in the second variable -which is the case of Nicholsontype systems, for example. Our results extend and improve some recent conclusions in the literature [2,3,9,[16][17][18], which mostly deal with scalar DDEs and/or cooperative ndimensional models. Special attention will be given to the case where each component f i of f = ( f 1 , .…”

“…In some contexts, they are interpreted as structured models for populations distributed over n different classes or patches, with migration among the patches, where x i (t) is the density of the species on class i, a ij (t) (j = i) is the migration coefficient from class j to class i, d i (t) the coefficient of instantaneous loss for class i, and f i (t, φ) is the growth function for class i. DDEs where the delays intervene in the linear terms have deserved the attention of a number of researchers, e.g. as patch structured population or SIS (susceptible-infective-susceptible) multi-strain epidemic models with time delays for the dispersal among patches [9,11,17]. We also refer the reader to [1][2][3]15,19], for real interpretation of the DDEs under consideration and more applications.…”

“…In the study of stability for nonautonous DDEs, a condition as (H2*) with v = 1 has been often presented (see e.g., [5,9]) in the equivalent form (for a positive denominator) lim inf t→∞…”

“…A number of methods has been proposed to tackle different situations, depending on whether the equations are autonomous or not, scalar or multi-dimensional, monotone or nonmonotone. See [1][2][3][4][5][6][7][8][9][10][11] and references therein for an explanation of the models and motivation from real world applications.…”

“…The criteria for permanence in [2,4,6] and other works demand that all coefficients are bounded. More recently, some authors have relaxed this restriction [8,9,15,16], though they still imposed some boundedness requirements. Here, the boundedness of all coefficients in (1) will not be a priori assumed.…”