Boundedness and persistence of delay differential equations with mixed nonlinearity

Abstract

For a nonlinear equation with several variable delayṡwhere the functions f k increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on [0, ∞), as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide.

Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications

“…A large-scale literature on the scalar Nicholson's blowflies equation, on a number of generalizations and on related models has been produced since its introduction by Gurney et al [13], and real world applications implemented. Nevertheless, a number of problems regarding scalar Nicholson-type equation still remain unsolved, see [2,3] and references therein. On the other hand, results concerning multi-dimensional versions of such models are still quite limited.…”

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

“…Permanence of solutions of a differential equation model is especially important in mathematical biology and ecology [7,16,25]. Permanence of solutions of scalar delay population models was recently studied in [1,2,6,14,16]. In [21] we considered a scalar delay population model…”

Persistence of nonautonomous logistic system with time-varying delays and impulsive perturbations

“…Let a = 0 or τ = 0 in Lemma 2, then one may derive Corollaries 1 and 2 easily. Since systems (15) and (16) include systems (2) and (3) as the special cases, Corollaries 1 and 2 are also suitable for systems (2) and 3, respectively. In the following, we will give our main result for the persistence of system (5).…”

“…In recent years, for modelling the dynamics of some biological populations, several delay differential systems of logistic type have been proposed and studied by many authors; see [2,4,5,7,9,13,19,29,31]. The classical logistic system with time delay can be described as follows:…”