# 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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**Sample text**

For a l l < ~ . j r. In p a r t i c u l a r , i f Moreover, f o r any n b( R Then equation. 2) has a periodic solution for all large enough r . The point of t h i s r e s u l t is that the above type of negative feedback delay can cause the b i f u r c a t i o n of a periodic solution which persists f o r a l l large enough delay. I t is yet not c l e a r whether such delays are responsible f o r biochemical oscillations. The estimates that Mahaffy has made from a v a i l a b l e data seem to i n d i - cate that the delays that are necessary to produce such o s c i l l a t i o n s may be too large to be o f b i o l o g i c a l importance, and other mechanisms may need to be found to explain such o s c i l l a t i o n s .

LO: 13-32. N. C. (1982). "On the use of reducible functional d i f f e r e n t i a l equations in biological models", J. Math Anal. App. 89: 46-66. -N. K. (1982). Methods of Bifurcation Theor7, Springer Verlag, New York. L. (1967). P. LaSalle, editors, Academic Press,New York: 167-183. L. and Yorke, J. (1973). "Some equations modelling growth processes and gonorrhea epidemics", Math. Biosciences, 16: 75-101. G. and Stephanopoulos, G. (1981). Science 213: 972-979. [13] Frazer, A. and T i v a r i , J.

1977). "A global s t a b i l i t y criterion for simple loops," J. Math. Biol. 4: 363-373. [2] An der Heiden, U. (1979). "Periodic solutions of a nonlinear second order d i f f e r e n t i a l equation with delay", J. Math. Anal. App. 70: 599-609. [3] Atkin, E. (1983). "Hopf bifurcation in the two locus genetic model", Mem. AMS 284, American Mathematical Society, Providence, RI. T. M. (1978). "Global asymptotic s t a b i l i t y of certain models for protein synthesis and repression", Quart.