Time scales in linear delayed differential equations

Sánchez, Esteban Morcillo; Parra, Rafael; Auger, Pierre; Gómez-Mourelo, Pablo

doi:10.1016/j.jmaa.2005.10.074

Cited by 10 publications

(8 citation statements)

References 20 publications

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“…Paper [2] (cf. [3,27,32]) presents a model of dynamics of a fish population with both age and vertical structures. The fish habitat is divided into N spatial patches and the fish densities n i in the ith patch satisfy the following system of equations: (16) …”

Section: 3mentioning

confidence: 99%

Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

Bobrowski

Bogucki²

2008

Studia Math.

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Abstract. Let S be a locally compact Hausdorff space. Let Ai, i = 0, 1, . . . , N , be generators of Feller semigroups in C0(S) with related Feller processes Xi = {Xi(t), t ≥ 0} and let αi, i = 0, . . . , N , be non-negative continuous functions on S withA natural interpretation of a related Feller process X = {X(t), t ≥ 0} is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ S, with probability αi(p), the process behaves like Xi, i = 0, 1, . . . , N. We provide an approximation of {T (t), t ≥ 0} via a sequence of semigroups acting in the Cartesian product of N + 1 copies of C0(S) that supports this interpretation, thus generalizing the main theorem of Bobrowski [J. Evolution Equations 7 (2007)] where the case N = 1 is treated. The result is motivated by examples from mathematical biology involving models of gene expression, gene regulation and fish dynamics.

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Section: 3mentioning

confidence: 99%

Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

Bobrowski

Bogucki²

2008

Studia Math.

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“…est très rapide en comparaison à l'autre : on parlera de dynamique lente et de dynamique rapide. Nous allons présenter les résultats relatifs au cas autonome développé dans [23,24].…”

Section: Méthodes D'agrégation Des Variables En Temps Discretunclassified

“…De telles méthodes ont d'abord été présentées dans [2] et ont été développées initialement pour les Equations aux Dérivées Ordinaires dans [5], [4] et [21]. Ces travaux ont ensuite été étendus aux modèles discrets ( [23] et [8]), et enfin aux Equations aux Dérivées Partielles [1,12] et Equations Différentielles à Retard [24]. Les idées derrière les méthodes d'agrégation des variables sont souvent intuitives et sont parfois utilisées de manière implicite, par exemple en épidémiologie où la dynamique des populations est ignorée car la dynamique épidémiologique est beaucoup plus rapide.…”

Section: Introductionunclassified

Applications de méthodes d’agrégation de variables à l’analyse de modèles spatiaux de dynamique des populations

Nguyen-Huu

Auger

2008

Revue Africaine De Recherche en Informatique Et Mathématiques Appliquées

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International audience Models in population dynamics can deal with an important number of parameters and variables, which can make them difficult to analyse. Aggregation of variables allow reducing complexity of such models by building simplified models governing fewer variables by use of the existence of different time scales associated to the processes governing the whole system. Those reduced models allows analysing and describing the global dynamics of the system. We present those methods for time discrete models and illustrate their use for the study of spatial host-parasitoids models. Les modèles de dynamique de populations peuvent prendre en compte un nombre important de paramètres et de variables, ce qui les rend difficiles à analyser. Lorsqu’il existe des processus associés à deux échelles de temps différentes, une lente et une rapide, les méthodes d’agrégation de variables permettent de construire un modèle simplifié qui comporte un nombre plus faible de variables. Elles permettent ainsi d’analyser et de décrire un système de manière globale. Nous présentons ces méthodes dans le cas de modèles discrets, puis nous illustrons leur utilisation à l’aide de modèles hôte-parasitoïdes spatialisés.

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“…For this reason, it is customary to write the argument of φ as a, 'a' standing for 'age. ' A model of dynamics of a Solea solea or Engraulis encrasicholus population with both age and vertical structures, due to Arino et al [1,20], uses the McKendrick model as a building block. In the model, the fish habitat is divided into N spatial patches and the fish densities, or age profiles φ i , in the ith patch satisfy the following system of equations:…”

Section: Introductionmentioning

confidence: 99%

“…In the context of Theorem 2.1, we take B = X × X, Q = Q, and then (20) follows by (8) with P = P. Also, B is the space X of vectors of the form αx βx , x ∈ X. Let C be the operator in B given by…”

mentioning

confidence: 99%

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

Banasiak

Bobrowski

2014

J. Evol. Equ.

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Abstract. Let α be a bounded linear operator in a Banach space X, and let A be a closed operator in this space. Suppose that for 1 , 2 mapping D(A) to another Banach space Y, A | ker 1 and A | ker 2 are generators of strongly continuous semigroups in X. Assume finally that A | ker a , where a = 1 α + 2 β and β = I X − α, is a generator also. In the case where X is an L 1 -type space, and α is an operator of multiplication by a function 0 ≤ α ≤ 1, it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of A | ker 1 with probability α(x) and is subject to the rules of A | ker 2 with probability β(x) = 1 − α(x). We provide an approximation (a singular perturbation) of the semigroup generated by A | ker a by semigroups built from those generated by A | ker 1 and A | ker 2 that supports this intuition.

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Time scales in linear delayed differential equations

Cited by 10 publications

References 20 publications

Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

Applications de méthodes d’agrégation de variables à l’analyse de modèles spatiaux de dynamique des populations

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

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