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PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic

Benoît Perthame (2004)

Applications of Mathematics

Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on...

Periodic dynamics in a model of immune system

Marek Bodnar, Urszula Foryś (2000)

Applicationes Mathematicae

The aim of this paper is to study periodic solutions of Marchuk's model, i.e. the system of ordinary differential equations with time delay describing the immune reactions. The Hopf bifurcation theorem is used to show the existence of a periodic solution for some values of the delay. Periodic dynamics caused by periodic immune reactivity or periodic initial data functions are compared. Autocorrelation functions are used to check the periodicity or quasiperiodicity of behaviour.

Periodic solution to a multispecies predator-prey competition dynamic system with Beddington-DeAngelis functional response and time delay

Xiaojie Lin, Zengji Du, Yansen Lv (2013)

Applications of Mathematics

In this paper, we are concerned with a delayed multispecies competition predator-prey dynamic system with Beddington-DeAngelis functional response. Some sufficient conditions which guarantee the existence of a positive periodic solution for the system are obtained by applying the Mawhin coincidence theory. The interesting thing is that the result is related to the delays, which is different from the corresponding ones known from literature (the results are delay-independent).

Periodic Solutions in a Mathematical Model for the Treatment of Chronic Myelogenous Leukemia

A. Halanay (2012)

Mathematical Modelling of Natural Phenomena

Existence and stability of periodic solutions are studied for a system of delay differential equations with two delays, with periodic coefficients. It models the evolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenous leukemia under a periodic treatment that acts only on mature cells. Existence of a guiding function leads to the proof of the existence of a strictly positive periodic solution by a theorem of Krasnoselskii....

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