Persistence and oscillations in the delay logistic equation with variable parameters (Q2769743)

The authors consider the logistic equation of the form NEWLINE\[NEWLINE\frac{dx(t)}{dt}=r(t)x(t) \Biggl[ 1-\frac{x(t-\tau(t))}{K(t)} \Biggr],\tag{*} NEWLINE\]NEWLINE where \(r,K,\tau\) are continuous functions with positive upper bound and lower bound, respectively. An estimation \(0<x_*\leq x(t)\leq x^*\) for \(t\geq T^*\) on solutions to (*) is given. For any two positive solutions \(x\) and \(y\) to (*), if \(\tau(t)<1\), a sufficient condition for \(x-y\) oscillating about zero is also obtained: the equation \(\lambda+\frac{r_L}{K_M}x_*e^{-\lambda \tau_L}=0\) has no real roots, with \( r_L=\inf_{t\in [0,\infty)}r(t)\), \(K_M=\sup_{t\in [0,\infty)}K(t),\;\tau_L= \inf_{t\in [0,\infty)}\tau(t)\). The above results are the trivial generalization of conclusions on the equation \(\frac{dx}{dt}=rx(1-\frac{x}{K})\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00066].