By Kennet L. Cooke (auth.), Mimmo Iannelli (eds.)

K.L. Cooke: hold up differential equations.- J.M. Cushing: Volterra integrodifferential equations in inhabitants dynamics.- K.P. Hadeler: Diffusion equations in biology.- S. Hastings: a few mathematical difficulties coming up in neurobiology.- F.C. Hoppensteadt: Perturbation tools in biology.- S.O. Londen: vital equations of Volterra type.

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O. 2) is called stable globally in the delay i f it is stable for each r of these concepts may be explained as follows. In the first place, in any model we do not know the delays exactly, but only in [O,cr». The relevance appro~mately. Therefore, we would like stability results to be unchanged by small changes in the delays, or we would like to discover values of the delay at which stability properties change. Secondly, if an equation is stable globally in the delay, then we know that no amount of delay introduced can destabilize the system.

And A A = b CP Ab , Q A ~ = 4> ~ Q A P _ 4> = <'I' A '~ > A i s in QA' be a finit e set of eigenvalues of {AI" " ,AN} the closed linear extension of the spaces MA(A) the generalized eigenspace associated with 11.. be the basis for sion n x d j where Si mila r l y , l et space associated with 'I' be the bas is for P* . II. 19) of and call be PA(A) d. x then cp N has dimen - J p* be the gen erali zed eigen- II. , For any real number A with real parts n I. 2. PA(A) [~;j 'I'N <'I' ,CP> Then Let Let and the formal adjoint II.

E. I CHAPTER III. STABILITY CONDITIONS FOR EXPONENTIAL POLYNOMIALS AND ENTIRE FUNCTIONS 10 . Criteria for stability. In the study of the linear autonomous system x(t) = L(x t), we have seen that the spectral properties are determined by the zeros of the characteristic function det~(A), the function ~(A) where if is ~(A) AI - o f -r e AS dn(8) • 46 If all zeros satisfy ReA < 0, then the zero solution of the FDE is exponen- tially asymptotically stable. Also, as we saw in Section 9, stability of an equilibrium solution of an autonomous nonlinear FDE is often studied by linearizing and trying to determine stability of the zero solution of the linear approximation.