Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost-periodic Nicholson systems

Abstract

Using techniques of non-autonomous dynamical systems, we completely characterize the persistence properties of an almost periodic Nicholson system in terms of some numerically computable exponents. Although similar results hold for a class of cooperative and sublinear models, in the general nonautonomous setting one has to consider persistence as a collective property of the family of systems over the hull: the reason is that uniform persistence is not a robust property in models given by almost periodic differential equations.Key words and phrases. Non-autonomous dynamical systems, almost periodic Nicholson models, uniform and strict persistence, mathematical biology.

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

“…We first apply our results to generalized Nicholson and Mackey-Glass systems. The literature is recent, though extensive, and here we mention a few selected references dealing with the persistence and permanence for multidimensional Nicholson equations [3,6,7,10,18] and Mackey-Glass equations [2,6,15], and references therein. Meaningful applications were given in the early work of Berezansky et al [3], where planar Nicholson systems were proposed as a compartment model for leukemia (with a delay representing the time required for the response of leukemia cells to growth) and a marine model with fishing and protected areas (with a maturation delay for the species).…”

“…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.…”

Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations

“…Under these assumptions, the existence of a continuous decomposition X = E(ω) ⊕ F (ω) is proved, where E(ω) is the principal Floquet subspace and the semiflow exhibits an exponential separation on the sum, but now F (ω) ∩ X + is not void and contains those positive vectors u satisfying U ω (t 1 ) u = 0. This dynamical behaviour is called exponential separation (or continuous separation) of type II, implicity referring to the classical concept as exponential separation of type I. Novo et al [24], Calzada et al [5] and Obaya and Sanz [25,26] show the importance of the exponential separation of type II in the study of linear and nonlinear nonautonomous functional differential equations with finite delay.…”

Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay

“…Later, Novo et al [11] introduced a variation of this notion, and called it continuous separation of type II, in order to make it applicable to delay equations. The results in Novo et al [12] and Obaya and Sanz [14,15] show its importance in the dynamical description of non-autonomous FDEs with finite delay, and now it becomes relevant in reaction-diffusion systems with delay.…”

“…In [15] Obaya and Sanz showed that in the general non-autonomous setting, in order that persistence can be detected experimentally, this notion should be considered as a collective property of the complete family of systems over Ω. We follow this collective approach to develop dynamical properties of persistence with important practical implications.…”