Dynamics of a delay logistic equation with slowly varying coefficients (Q2420920)

In this paper, the dynamics of the logistic delay differential equation \[\frac{du}{dt} = r(\varepsilon t)[1-a(\varepsilon t)u(t-T(\varepsilon t))]u(t)\] is investigated, where $\varepsilon\in (0, 1)$ is a very small parameter and $a(\tau), r(\tau), T(\tau)$ are positive trigonometric polynomials. It shows that every positive solution is ultimately bounded by a number $M>0$ in terms of $a, r$ and $T$. Moreover, for each sufficiently small $\varepsilon$, the equation has a solution of the form \[u_0(\tau, \varepsilon) = \frac{1}{a(\tau)}+\sum^{\infty}_{i=1}\varepsilon^iu_i(\tau)\] bounded for $\tau\in (-\infty, \infty)$. Then, the stability of $u_0(\tau, \varepsilon)$ in a nearly critical case is analysed and Andronov-Hopf bifurcation is also considered. Under further assumptions, an asymptotic expansion is constructed for a regular almost periodic solution. It is clear that the dynamics of the above equation is much more complicated than that for logistic equations with constant coefficients.