Periodic solutions for small and large delays in a tumor-immune system model.
Yafia, Radouane (2006)
Electronic Journal of Differential Equations (EJDE) [electronic only]
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Yafia, Radouane (2006)
Electronic Journal of Differential Equations (EJDE) [electronic only]
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Banibrata Mukhopadhyay, Rakhi Bhattacharyya (2006)
Applications of Mathematics
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In the present paper, a mathematical model, originally proposed by Danziger and Elmergreen and describing the thyroid-pituitary homeostatic mechanism, is modified and analyzed for its physiological and clinical significance. The influence of different system parameters on the stability behavior of the system is discussed. The transportation delays of different hormones in the bloodstream, both in the discrete and distributed forms, are considered. Delayed models are analyzed regarding...
Radouane Yafia (2009)
Applicationes Mathematicae
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We consider a system of delay differential equations modelling the tumor-immune system competition with negative immune response and three positive stationary points. The dynamics of the first two positive solutions are studied in terms of the local stability. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence and stability of a limit cycle bifurcating from the second positive stationary point, when the delay (taken as a parameter) crosses...
Urszula Foryś, Mikhail Kolev (2003)
Banach Center Publications
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The role of time delays in solid avascular tumour growth is considered. The model is formulated in terms of a reaction-diffusion equation and mass conservation law. Two main processes are taken into account-proliferation and apoptosis. We introduce time delay first in underlying apoptosis only and then in both processes. In the absence of necrosis the model reduces to one ordinary differential equation with one discrete delay which describes the changes of tumour radius. Basic properties...
Fofana, M.S. (2005)
International Journal of Mathematics and Mathematical Sciences
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Sanyi Tang (2010)
ESAIM: Mathematical Modelling and Numerical Analysis
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In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation...
Banibrata Mukhopadhyay, Rakhi Bhattacharyya (2010)
Applications of Mathematics
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We consider a mathematical model of nutrient-autotroph-herbivore interaction with nutrient recycling from both autotroph and herbivore. Local and global stability criteria of the model are studied in terms of system parameters. Next we incorporate the time required for recycling of nutrient from herbivore as a constant discrete time delay. The resulting DDE model is analyzed regarding stability and bifurcation aspects. Finally, we assume the recycling delay in the oscillatory form to...
Ye Cheng, Bao Shi, Liangliang Ding (2021)
Applications of Mathematics
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To explore the impacts of time delay on nonlinear dynamics of consensus models, we incorporate time-varying delay into a two-agent system to study its long-time behaviors. By the classical 3/2 stability theory, we establish a sufficient condition for the system to experience unconditional consensus. Numerical examples show the effectiveness of the proposed protocols and present possible Hopf bifurcations when the time delay changes.
Gazi, Nurul Huda, Bandyopadhyay, Malay (2006)
International Journal of Mathematics and Mathematical Sciences
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Eva Sánchez (2003)
Banach Center Publications
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This paper provides an introduction to delay differential equations together with a short survey on state-dependent delay differential equations arising in population dynamics. Our main goal is to examine how the delays emerge from inner mechanisms in the model, how they induce oscillations and stability switches in the system and how the qualitative behaviour of a biological model depends on the form of the delay.