Delayed logistic population models revisited
Consider the blowfly logistic equation \[ x'(t)=-\delta x(t)+rx(t-\tau)\max\{1-x(t-\tau), 0\}, \] where \(r\) and \(\delta\) are constants. In the case \(1<\frac{r}{\delta}<4\), the author proves that the positive equilibrium \(K_{\delta}=1-\frac{\delta}{r}\) of the considered equation is globally asymptotically stable if either \(p\leq 2\) or \(p>2\) and \(0<\tau<\frac{1}{\delta}\log\left(\frac{p-1}{p-2}\right)\). The author discusses the global behaviour of the solutions as the parameter \(\delta\) is increased. It is not observed any complex behaviour for a small value of \(\tau\). If the positive equilibrium is asymptotically stable, then it seems to be globally attracting. If the positive equilibrium is unstable, then there is a simple attracting nontrivial periodic solution. The bifurcation diagram is similar to a bubble and a stable positive equilibrium cannot be destabilized increasing \(\delta\). For small values of \(\delta\), there are sustained oscillations. The amplitude of the periodic solutions is not a monotone decreasing function of \(\delta\). The average value of the solutions can be larger than the positive equilibrium \(K_{\delta}\).
logistic models delay-differential equations
scientific article; zbMATH DE number 6381937