# zbMATH — the first resource for mathematics

Bifurcation analysis on a survival red blood cells model. (English) Zbl 1105.34047
The paper considers a scalar delay differential equation of the form $\dot u (t)=-\sigma u(t)+\exp(-u(t-\nu)),$ depending on the two positive parameters $$\sigma$$ and $$\nu$$. First, the authors find a sequence of critical values $$\nu_k$$ for which the unique equilibrium undergoes a supercritical Hopf bifurcation.
Then, the authors apply J. Wu’s degree theoretical result [Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)] to show that the periodic solutions emanating from the Hopf bifurcation continue to exist for all $$\nu\in(\nu_k,\infty)$$. Having the paper of Wu at hand is recommended because the notation in the degree theoretical part of the paper is not self-contained.

##### MSC:
 34K18 Bifurcation theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations 34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text:
##### References:
 [1] Chow, S.N., Existence of periodic solutions of autonomous functional differential equations, J. differential equations, 15, 350-378, (1974) · Zbl 0295.34055 [2] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations and applications to bogdanov – takens singularity, J. differential equations, 122, 201-224, (1995) · Zbl 0836.34069 [3] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations, 122, 181-200, (1995) · Zbl 0836.34068 [4] Gopalsamy, K.; Trofimchuk, S.I., Almost periodic solutions of lasota – wazewska-type delay differential equation, J. math. anal. appl., 237, 106-127, (1999) · Zbl 0936.34058 [5] Györi, I.; Ladas, G., Oscillation theory of delay differential equations with applications, (1991), Clarendon Oxford · Zbl 0780.34048 [6] Karakostas, G.; Philos, C.G.; Sficas, Y.G., Stable steady state of some population models, J. dynam. differential equations, 4, 161-190, (1992) · Zbl 0744.34071 [7] Kulenovic, M.R.S.; Ladas, G.; Sficas, Y.S., Global attractivity in population dynamics, Comput. math. appl., 18, 925-928, (1989) · Zbl 0686.92019 [8] Li, J.W.; Cheng, S.S., Global attractivity in an RBC survival model of wazewska and lasota, Quart. appl. math., 60, 477-483, (2002) · Zbl 1022.92008 [9] Ruan, S.G.; Wei, J.J., Periodic solutions of planar systems with two delays, Proc. roy. soc. Edinburgh sect. A, 129, 1017-1032, (1999) · Zbl 0946.34062 [10] Y.L. Song, Positive periodic solution of a periodic survival red blood cells model, Appl. Anal., in press [11] Wazewska-Czyzewska, M.; Lasota, A., Mathematical problems of the dynamics of the dynamics of the red blood cells system, Mat. stos. (3), 6, 23-40, (1976) [12] Wei, J.J.; Li, M.Y., Hopf bifurcation analysis for a delayed Nicholson blowflies equation, Nonlinear anal., 60, 1351-1367, (2005) · Zbl 1144.34373 [13] Wu, J.H., Symmetric functional differential equations and neural networks with memory, Trans. amer. math. soc., 350, 4799-4838, (1998) · Zbl 0905.34034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.