Global positive solutions of a generalized logistic equation with bounded and unbounded coefficients

In this paper we study the generalized logistic equation $$ frac{du}{dt}=a(t)u^{n}-b(t)u^{n+(2k+1)},quad n,kin mathbb{N}, $$ which governs the population growth of a self-limiting specie, with $a(t)$, $b(t)$ being continuous bounded functions. We obtain a unique global, positive and bounded solution which, further, plays the role of a frontier which clarifies the asymptotic behavior or extensibility backwards and further it is an attractor forward of all positive solutions. We prove also that the function $$ phi (t)=sqrt[2k+1]{a(t)/b(t)} $$ plays a fundamental role in the study of logistic equations since if it is monotone, then it is an attractor of positive solutions forward in time. Furthermore, we may relax the boundeness assumption on $a(t)$ and $b(t)$ to a boundeness of it. An existence result of a positive periodic solution is also given for the case where $a(t)$ and $b(t)$ are also periodic (actually we derive a necessary and sufficient condition for that). Our technique is a topological one of Knesser's type (connecteness and compactness of the solutions funnel)