STABILITY OF LIMIT CYCLE IN A DELAYED MODEL FOR TUMOR IMMUNE SYSTEM COMPETITION WITH NEGATIVE IMMUNE RESPONSE

This paper is devoted to the study of the stability of limit cycles of a system of nonlinear delay differential equations with a discrete delay. The system arises from a model of population dynamics describing the competition between tumor and immune system with negative immune response. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions, using the methods presented by Diekmann et al

Periodic solutions for small and large delays in a tumor-immune system model

In this paper we study the Hopf bifurcation for the tumor-immune system model with one delay. This model is governed by a system of two differential equations with one delay. We show that the system may have periodic solutions for small and large delays for some critical value of the delay parameter via Hopf bifurcation theorem bifurcating from the non trivial steady state

Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response

This paper is devoted to the study of the stability of limit cycles of a system of nonlinear delay differential equations with a discrete delay. The system arises from a model of population dynamics describing the competition between tumor and immune system with negative immune response. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions, using the methods presented by Diekmann et al

Contribution à l'Etude de la Bifurcation de Hopf dans le Cadre des Equations Différentielles à Retard, Application à un Problème en Dynamique de Population.

Our first goal in this work is to give a proof of exchange of stability from the trivialbranch to the bifurcated one. This proof is based on the two following steps:i) reduction of the equation to a two-dimensional system via the variation of constantformula end the center manifold theorem.ii) Estimation of the distance between solutions of the original equation and the bifurcatedperiodic solutions.We obtain an estimate of the stability region.The second goal is to study the dynamics of Haematopoietic Stem Cells (HSC) Modelwith one delay.The model, was initially introduced by Mackey (1978). There are two possible stationarystates. One of them is trivial and unstable, the second is nontrivial, depending onthe delay \tau.We prove the existence of a critical value ¿0 of the delay \tau in which the exchange ofstability of nontrivial stationary state may occur.We introduce also an approachable model depending on this critical value of the delay,such that the nontrivial stationary state do not depend on the delay which is the sameone of Mackey model at \tau =\tau_{0}.By a similar study of the approachable model as in Mackey model, we obtain the existenceof the bifurcated periodic solution branch around the nontrivial stationary state.In the end, we give an explicit algorithm for calculating the elements of bifurcation.Notre premier objectif dans ce travail est de donner une démonstration du changementde la stabilité de la branche supercritique de solutions périodiques bifurquéesdans le cadre des équations diérentielles à retard, en se basant sur les deux étapessuivantes:(i) Réduction de l'équation à un système en dimension deux par la formule de variationde la constante et le théorème de la variété centre.(ii) Estimation de la distance entre la solution de l'équation initiale et la solution pé-riodique bifurquée.Nous obtenons ainsi un domaine de stabilité de la branche supercritique.Le second objectif est d'étudier une équation différentielle à un seul retard issued'un modèle en dynamique de population cellulaire sanguine (Haematopoiese).Ce modèle, initialement introduit par Mackey (1978) présente une position d'équilibretriviale qui est instable et une famille de positions d'équilibre non triviales dont lastabilité dépend du retard.Nous montrons l'existence d'une valeur critique ¿0 du retard \tau autour de laquelle nousobtenons un changement de stabilité de cette famille de positions d'équilibre en fonctiondu retard.Nous avons ainsi introduit un modèle approché en fonction de cette valeur critique duretard qui coincide avec celui de Mackey pour la valeur du retard \tau = \tau_{0}. Le modèleapproché possède un point d'équilibre trivial et un non trivial ne dépendant pas duretard.Par une étude du modèle approché analogue à celle du modèle de Mackey, nous obtenonsen particulier l'existence d'une branche de solutions périodiques bifurquées àpartir du point d'équilibre non trivial. Enn nous donnons un algorithme explicite decalcul des éléments de la bifurcation

Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response

The dynamics of the model for tumor-immune system competition with negative immune response and with one delay investigated. We show that the asymptotic behavior depends crucially on the time delay parameter. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state, by using the delay as a parameter of bifurcation. The obtained results provide the oscillations given by the numerical study in M. Gałach (2003), which are observed in reality by Kirschner and Panetta (1998)

Modeling and Dynamics of Predator Prey Systems on a Circular Domain

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