yongwimon lenbury department of mathematics, faculty of science, mahidol university
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Mahidol University. Wisdom of the Land. Dynamical Analysis of Nonlinear Delayed Differential Equation Models. Yongwimon Lenbury Department of Mathematics, Faculty of Science, Mahidol University Centre of Excellence in Mathematics, PERDO, Commission on Higher Education Thailand. C. M. E. - PowerPoint PPT PresentationTRANSCRIPT
Yongwimon LenburyYongwimon LenburyDepartment of Mathematics, Department of Mathematics, Faculty of Science, Mahidol Faculty of Science, Mahidol
UniversityUniversityCentre of Excellence in Centre of Excellence in
Mathematics, PERDO, Commission Mathematics, PERDO, Commission on Higher Educationon Higher Education
ThailandThailand
C
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Analysis of 2D delayed models using omega limit sets and full time solutions: insulin-glucose model with delays.
One dimensional nonlinear population delayed model: construction of full time solutions
Introduction
Conclusion
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Analysis of model systems with delays may be done by applying the Hopf bifurcation theory. It is also possible, to make use of the concept of
- limit sets and the construction of full time solutions. After a brief introduction of the full time solutions, we illustrate their use in the analysis of one dimensional delayed nonlinear population model, in which the growth function needs not be monotone, and a two dimensional model of insulin-glucose system with delays.
Many systems exhibits delays in responses to stimuli, e.g. insulin-glucose control system.
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I.
(D. V. Giang, Y. Lenbury, Seidman, 2007)
Given a
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II.
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(Giang, Lenbury, De Geatano, 2008)
The ability to maintain a relatively constant glucose concentration is an essential feature of life. The
hormone mainly responsible for the maintainance of blood glucoe levels is insulin, secreted by the pancreas. We consider the following model for the glucose G(t)
(mM) and the insulin I(t) (pM) proposed by Pulumbo et al. (2007).
( ) ( ) ( ) ( ) (3.1)ghxg xgi i
G
TG t K G t K G t I t
V
max( ) ( ) ( ( )) (3.2)iGxi g
I
TI t K I t f G t
V
0*
( ) , with ( ) 1.supG
Gf G f G
G G
III.
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of (3.1)- (3.2)
4.Let (Gb, Ib) be equilibrium point satisfying
IV.
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5.
(3.1)- (3.2)
V.
,
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satisfying
5.VI.
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(3.1)-(3.2)
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t0 t* t0 +2 t1 t
max I
0 2 * is .t t I fastly oscillated
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(6.5)
(6.6)
Since I has a maximum at t = 0,
from which it can be shown that
We then have
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(5.1).
(6.5) (6.6) is similar.
By putting in (6.1) we havegt
integrating (6.2) both sides we get
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VII.
We have illustrated how the use of omega limit sets and full time solutions may be more appropriate and more illuminating than the application of the more conventional
methods, such as the Hopf bifurcation, or singular perturbation analysis, with which only local asymptotic behavior or
stability may be established. The Lyapunov functions, on the other hand may give information on global stability but cannot
easily be derived for most model equations.
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This work has been supported by the National Centre for Genetic Engineering and Biotechnology, and the Centre of Excellence in
Mathematics, Thailand.
Thank YouThank You
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http://cem.sc.mahidol.ac.th/ConferenceWeb/index.html
17-19 DEC, 2009Twin Towers Hotel, Bangkok