By Stavros N. Busenberg (auth.), Luigi Salvadori (eds.)
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Additional resources for Bifurcation Theory and Applications: Lectures given at the 2nd 1983 Session of the Centro Internationale Matematico Estivo (C.I.M.E.) held at Montecatini, Italy, June 24 – July 2, 1983
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.  MacDonald, N. (1978). Time Lag in Biological Models, Lecture Notes in Biomathematics, Vol. 27, Springer, New York.  Monod, J. (1942). Researches sur la Croissence des Cultures Bacteriennes, Hermann, Pari s.  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  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.  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.  Green, D. (1978). "Self-oscillations for epidemic models", Math. Biosciences 38: 91-111.  Grossman,Z. and Gumowski, I. (1976). "Self-sustained oscillations in the Jacob-Monod model of gene regulation", Lecture notes in Computer Science, Vol.