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Bifurcation Theory and Applications: Lectures given at the by Stavros N. Busenberg (auth.), Luigi Salvadori (eds.)

By Stavros N. Busenberg (auth.), Luigi Salvadori (eds.)

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Periodic solutions of certain protein synthesis models", J. Math. Anal. Appl. 74: 72-I05. E. and MacCracken, M. (1976). The Hopf Bifurcation and i t s Applications, Springer, New York. [29] MacDonald, N. (1978). Time Lag in Biological Models, Lecture Notes in Biomathematics, Vol. 27, Springer, New York. [30] Monod, J. (1942). Researches sur la Croissence des Cultures Bacteriennes, Hermann, Pari s. [31] Negrini, P. and Salvadori, L. (1979). " A t t r a c t i v i t y and Hopf Bifurcation", Nonlinear Analysis 3: 87-99.

Moreover, the process of transcription and translation of the genetic information carried by the DNA molecule takes a considerable amount of time and is another source of delays. So, a time delay is introduced in order to take into account some of the effects of this situation. 1) dYi(t) dt - aiYi-l(t-Ti) - biYi(t)' i = 2,3 . . . n, Ti > 0. An der Heiden [2] showed that, i f one makes a simple substitution, one can always reduce such a model to an equivalent one with only one delay. 1) setting x =y - y Tl = r and Ti = 0.

Biosciences, 16: 75-101. G. and Stephanopoulos, G. (1981). Science 213: 972-979. [13] Frazer, A. and T i v a r i , J. (1974). "Genetical feedback repression: genetic systems", J. Theor. Biol. 47: 397-412. C. (1963). "Oscillatory behavior of enzymatic control processes", Adv. Enzjzme Res. 3: 425-439. [15] Green, D. (1978). "Self-oscillations for epidemic models", Math. Biosciences 38: 91-111. [16] Grossman,Z. and Gumowski, I. (1976). "Self-sustained oscillations in the Jacob-Monod model of gene regulation", Lecture notes in Computer Science, Vol.

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