A unified approach to persistence

Hofbauer, Josef

doi:10.1007/bf00046670

Cited by 31 publications

(14 citation statements)

References 16 publications

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“…This imply that P is an average Lyapunov function (see e.g. [10], [11]) and using Th.12.2.2 from [11] it follows that E + 2 cannot be attained from int Ω 1 .…”

Section: Main Resultmentioning

confidence: 95%

Robust control for an uncertain chemostat model

Gouzé¹,

Robledo²

2005

Intl J Robust & Nonlinear

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Abstract. In this paper we consider a control problem for an uncertain chemostat model with a general growth function and cell mortality. This uncertainty affects the model (growth function) as well as the outputs (measurements of substrate). Despite this lack of information, an upper bound and a lower bound for those uncertainties are assumed to be known a priori. We build a family of feedback control laws on the dilution rate, giving a guaranteed estimation on the unmeasured variable (biomass), and stabilizing the two variables in a rectangular set, around a reference value of the substrate. We give two realistic applications of this control law to a depollution process and to phytoplankton culture.

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“…This imply that P is an average Lyapunov function (see e.g. [10], [11]) and using Th.12.2.2 from [11] it follows that E + 2 cannot be attained from int Ω 1 .…”

Section: Main Resultmentioning

confidence: 95%

Robust control for an uncertain chemostat model

Gouzé¹,

Robledo²

2005

Intl J Robust & Nonlinear

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“…Proposition 4 (See [16]). Let P an Average Lyapunov function and let Λ = {r i ∈ Γ : φ t (r i ) = r i for any t ∈ R} .…”

Section: Discussionmentioning

confidence: 99%

“…Definition 2 (See [16]). Let φ t be a semiflow defined in a compact metric space (X, d) and let Γ a closed and invariant subset of X.…”

Section: Appendixmentioning

confidence: 99%

Global stability for a model of competition in the chemostat with microbial inputs

Robledo¹,

Grognard²,

Gouzé³

2012

Nonlinear Analysis: Real World Applications

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We propose a model of competition of n species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n−1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input

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“…Theorem 4.2 may be viewed as a unified and generalized theorem combining results of Fonda (1988), Freedman and Moson (1990), Hofbauer (1989), andHotbauer and.…”

Section: (I) 3 R Is Uniformly Persistent (Ii) ~ Is Weakly Persistentmentioning

confidence: 99%

Uniform persistence and flows near a closed positively invariant set

1994

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In this paper, the behavior of a continuous flow in the vicinity of a closed positively .invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equiyalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider rarity of ecological models.

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A unified approach to persistence

Cited by 31 publications

References 16 publications

Robust control for an uncertain chemostat model

Robust control for an uncertain chemostat model

Global stability for a model of competition in the chemostat with microbial inputs

Uniform persistence and flows near a closed positively invariant set

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