1. Yongwimon Lenbury Department of Mathematics, Faculty of Science, Mahidol University Centre of Excellence in Mathematics, PERDO, Commission on Higher Education Thailand C M E

2. 2 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

3. 3 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.

4. 4

5. 5 I. (D. V. Giang, Y. Lenbury, Seidman, 2007) Given a

6. 6 II.

7. 7

8. 8 (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). III.

9. 9

10. 10

11. 11 of (3.1)- (3.2) 4. Let (G b, I b ) be equilibrium point satisfying IV.

12. 12 5. (3.1)- (3.2) V.,

13. 13 satisfying 5. VI.

14. 14 (3.1)-(3.2)

15. 15 t 0 t* t 0 + 2 t 1 t max I

16. 16 (6.5) (6.6) Since I has a maximum at t = 0, from which it can be shown that We then have

17. 17 (5.1). (6.5) (6.6) is similar. integrating (6.2) both sides we get

18. 18 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.

19. 19 This work has been supported by the National Centre for Genetic Engineering and Biotechnology, and the Centre of Excellence in Mathematics, Thailand. Thank You C M E

20. 20 http://cem.sc.mahidol.ac.th/ConferenceWeb/index.html 17-19 DEC, 2009 Twin Towers Hotel, Bangkok