Non-autonomous Functional Differential Equations and Applications

Novo, Sylvia; Obaya, Rafael

doi:10.1007/978-3-642-32906-7_4

Cited by 10 publications

(6 citation statements)

References 99 publications

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“…As a result, it is well-known that the semiflow is monotone (for instance see Novo et al [31]). Finally, it is also globally defined, as stated in the next result whose proof can be found in [29]. Proposition 6.3.…”

Section: Uniform Persistence In Monotone and Sublinear Concave Or Con...mentioning

confidence: 82%

“…) where (z m 0 ) k denotes, as usual, the k th component of the vector z m 0 0. Since 2 a i 1 k (ω 1 , x 1 ) (z m 0 ) k > 0, lemma 3.15 in [29] ensures that 0 is a strong sub-equilibrium for (5.11). Then, as we have mentioned before, there exists a t i 1 > 0 and a z 0i 1 > 0 such that, if h(t, ω, x, 0) is the solution of (5.11) with initial condition 0 ∈ R, then h(t, ω, x, 0) 0 for any t 0 and (ω, x) ∈ K, and besides h(t, ω, x, 0) > 2 z 0i 1 for any t t i 1 , (ω, x) ∈ K.…”

Section: Definition 52mentioning

confidence: 98%

“…This formulation allows us to apply techniques of ergodic theory, topological and differentiable dynamics to investigate the behaviour of the corresponding semiflow. The papers Chow and Leiva [3,4], Johnson and Mantellini [18], Johnson and Moser [19], Johnson et al [20], Huang and Yi [16], Sacker and Sell [39][40][41], Novo and Obaya [29], Novo et al [32,33], Poláčik and Tereščák [37], Shen and Yi [45] and Yi [52] can be seen as the basis for this work and contain the mathematical ingredients which will be used throughout this work.…”

Section: Introductionmentioning

confidence: 99%

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Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows

2013

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Several results of uniform persistence above and below a minimal set of an abstract monotone skew-product semiflow are obtained. When the minimal set has a continuous separation the results are given in terms of the principal spectrum. In the case that the semiflow is generated by the solutions of a family of non-autonomous differential equations of ordinary, delay or parabolic type, the former results are strongly improved. A method of calculus of the upper Lyapunov exponent of the minimal set is also determined.

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Section: Uniform Persistence In Monotone and Sublinear Concave Or Con...mentioning

confidence: 82%

Section: Definition 52mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows

2013

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“…In monotone non-autonomous dynamical systems, the so-called semiequilibria, introduced both for the deterministic and random cases (see [14] and [4]), are useful objects to determine invariant zones (see also Novo and Obaya [15], among others).…”

Section: Preliminaries On the Exponential Ordering And Some Results F...mentioning

confidence: 99%

The exponential ordering for nonautonomous delay systems with applications to compartmental Nicholson systems

Novo¹,

Obaya²,

Sanz³

et al. 2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

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“…In this situation 0 is a strong sub-equilibrium (see lemma 3.15 in [19], which applies to ODEs). As a consequence, there exist t i2 > 0 and m i2 > 0 such that, if h(t, ω, 0) is the solution of (3.10) with initial value 0, then h(t, ω, 0) > m i2 for any t t i2 and any ω ∈ Ω.…”

Section: Theorem 34 Let Us Consider the Almost Periodic Nicholson Sys...mentioning

confidence: 99%

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

Obaya

Sanz

2018

Nonlinearity

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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.

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Non-autonomous Functional Differential Equations and Applications

Cited by 10 publications

References 99 publications

Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows

Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows

The exponential ordering for nonautonomous delay systems with applications to compartmental Nicholson systems

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

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