In [3], [4] a mathematical model of chronic myelogenous leukemia is considered

Georgescu, R. M.

Ion, A. V.

English

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Simulation of the Draft Tube Vortex in Francis Turbines at Partial Discharge// Proceedings
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BAUTIN-TYPE BIFURCATIONS AND STABILITY OF EMERGED
SOLUTIONS FOR A DELAY DIFFERENTIAL EQUATION MODELING
LEUKEMIA
A. V. Ion, R. M. Georgescu (Bucharest, Romania)
In [3], [4] a mathematical model of chronic myelogenous leukemia is considered. Its study
may be reduced to that of the delay differential equation
x˙(t) = −[β(x(t)) + δ]x(t) + kβ(x(t− r))x(t− r), (1)
where β(x) = β0/(1 + xn), β0, n, δ, k, are positive parameters, and r > 0 is the delay. The
significance of x(·) and of parameters is presented in the above cited works, and a brief study
of equilibrium points and of their stability, as well as a comprehensive numerical exploration
of the solution and its dependence on the parameters is performed there.
In [2] we make an extensive study of the stability of the two equilibrium solutions,
completing (and, for a certain zone of the parameter space, correcting) the results of stability
from [3], [4]. We indicate in [2] the situations when Hopf bifurcations occur and we study
the orbital stability of the periodic solutions thus emerged, by computing the normal form
of the restriction of the equation to the bi-dimensional center manifold, and by determining
the sign of the first Lyapunov coefficient at the Hopf bifurcation point.
In the present work we show that this problem presents, for some values of the involved
parameters, Bautin-type bifurcations. This implies that, for some zone of the parameters
space, the problem has two limit cycles, one inside the other. Since the periodic solutions of
the problem are important, this mathematical result might be valuable for the biologists. We
present a numerical procedure to find Bautin-type bifurcation points. We study the stability
of the periodic orbits emerged by this bifurcation, by computing the second Lyapunov
coefficient at such a bifurcation point (by the method of [1]).
Acknowledgement. This work by is supported by the Russian Foundation for Basic
Research – Romanian Academy collaboration grant.
References
1. A.-V. Ion, An example of Bautin-type bifurcation in a delay differential equation// J.
Math. Anal. Appl. 2007. No. 329. P. 777-789.
2. A.-V. Ion, R.-M. Georgescu, Stability of equilibrium and periodic solutions of a delay
equation modeling leukemia, Proc. of “7th Workshop on Math. Modeling of Environmental
and Life Science Problems”, “Ovidius” University, Constant¸a, Romania, 25-28 October 2008,
in press.
3. L. Pujo-Menjouet, M. C. Mackey, Contribution to the study of periodic chronic myelogenous
leukemia// C. R. Biologies. 2004. Vol. 327. P. 235–244.
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Bautin-type bifurkations and stability of energy solutions for a delay differential equation modeling leukemia

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Bautin-type bifurkations and stability of energy solutions for a delay differential equation modeling leukemia

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