Abstract
In this paper, we obtain sufficient conditions for the persistence and permanence of a family of nonautonomous systems of delay differential equations. This family includes structured models from mathematical biology, with either discrete or distributed delays in both the linear and nonlinear terms, and where typically the nonlinear terms are nonmonotone. Applications to systems inspired by mathematical biology models are given.
Highlights
We investigate the persistence and permanence for a class of multidimensional nonautonomous delay differential equations (DDEs), which includes some structured models used in population dynamics, epidemiology, and other fields
We have proven the persistence and permanence of delayed differential systems (8) which incorporate distributed delays in both the linear and nonlinear parts and are in general noncooperative
The main theorem, Theorem 3, extends known results in recent literature [2,3,6,7,9,16,18], as it applies to a broad family of nonautonomous delay differential systems
Summary
We investigate the persistence and permanence for a class of multidimensional nonautonomous delay differential equations (DDEs), which includes some structured models used in population dynamics, epidemiology, and other fields. If the set C0+ is (positively) invariant for (1), the notions of (uniform) persistence, permanence, and stability always refer to solutions with initial conditions in C0+. In this way, we say that the system is uniformly persistent (in C0+) if there exists a positive uniform lower bound for all solutions with initial conditions in C0+, i.e., there is m > 0 such that all solutions x(t) = x(t, 0, φ) with φ ∈ C0+ defined on R+ and satisfy xi(t, 0, φ) ≥ m for t 1 and i = 1, . The paper ends with a short section of conclusions and open problems
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