Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

Abstract

Abstract. We determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of nonautonomous families of cooperative systems of ODEs or delay FDEs in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation. Our conditions are also necessary in the linear case. We apply our results to a noncooperative almost periodic Nicholson system with a patch structure, whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution.

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

“…As a consequence, there is a general need for definitions of persistence given globally for the family of systems over the hull of a particular non-autonomous system with a recurrent behaviour in time. This is the collective approach that has been taken in [22] and [26]. This supports the coherence of the results in the previous section, as what happens in Nicholson systems regarding persistence cannot at all be given for granted.…”

“…It is straightforward to check that, under hypothesis (a5), all the results in Section 6 in [26] still apply. For the sake of completeness we include here the following result, whose items are respectively Theorem 6.1 and Theorem 6.2 in [26].…”

“…In this situation, the properties of uniform persistence and strict persistence at 0 for the family of systems (3.2) have the following collective formulation, directly adapted from Definitions 3.1 and 5.2 in [26], respectively. .2) is uniformly persistent (u-persistent for short) if there exists a map ψ ≫ 0 such that for any ω ∈ Ω and any initial map ϕ ≫ 0 there exists a time t 0 = t 0 (ω, ϕ) such that y t (ω, ϕ) ≥ ψ for any t ≥ t 0 .…”

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

“…We follow this collective approach to develop dynamical properties of persistence with important practical implications. Our study intends to extend the theory of persistence written in Novo et al [12] and Obaya and Sanz [14] for non-autonomous ODEs, FDEs with delay and parabolic PDEs to parabolic PFDEs, considering also the cases of Robin or Dirichlet boundary conditions. We briefly explain the structure and contents of the paper.…”