Oscillatory Solutions for Certain Delay-differential Equations

The existence of oscillatory solutions for a certain class of scalar first order delay-differential equations is proved. An application to a delay logistic equation arising in certain models for population variation of a single specie in a constant environment with limited resources for growth is considered. It is known (cf. [1, 2]) that all solutions of the delay logistic equations (1) N'(t) = N(t)(a-bN(t)-N(t-l)), í > 0, with N(t) = No(t) > 0, — 1 < í < 0, N0 continuous, and a and 6 positive constants, satisfy 7V(i) —> a/(b+1) as t —► oo whenever b > 1. In [3] it was shown that for any b > 0, there exists a(b) > 0 and that if 0 < a < a(b), there exist solutions A^i) of (1) which do not oscillate about the equilibrium N = a/(b+ 1); in particular, such that, N(t) > a/(b + 1) for í > 0. It is the purpose of this paper to show that for this same a(b), if a > a(b), there exist oscillatory solutions about this equilibrium solution. In case b < 1, this is known; in fact, a Hopf bifurcation (cf. [1]) shows the existence for certain a of nonconstant positive periodic solutions. However, if b > 1, the fact that some solutions of (1) approach a/(b+1) in an oscillatory fashion seems to be new. The above mentioned result for (1) will follow from a result for a more general scalar delay-differential equation of the form (2) y'(t)=L(yt) + N(t,yt), Í > 0. Here yt = y(t + 6), —1 < 0 < 0, and we assume (Hi) L() is continuous and linear on C = C([-l, 0], R) and N(t, 4>) is continuous on R x C and satisfies \N(t, 0; where the norm in C is defined by ||<¿>|| = sup{|0(ö)| : -1 < 6 < 0}, and /0°° M(t) dt < oo; (H2) The characteristic equation for (3) y'(t) = L(yt) has a pair of simple pure imaginary roots ±iß, ß > 0, and all other roots have negative real parts. Received by the editors October 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 34K20.