Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 119, pp. 1-13.
Title: Global positive solutions of a generalized logistic
equation with bounded and unbounded coefficients
Authors: George N. Galanis (Naval Academy of Greece)
Panos K. Palamides (Naval Academy of Greece)
Abstract:
In this paper we study the generalized logistic equation
$$
\frac{du}{dt}=a(t)u^{n}-b(t)u^{n+(2k+1)},\quad n,k\in \mathbb{N},
$$
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).
Submitted October 13, 2003. Published December 1, 2003.
Math Subject Classifications: 34B18, 34A12, 34B15.
Key Words: Generalized logistic equation; asymptotic behavior
of solutions; periodic solutions; Knesser's property;
Consequent mapping; Continuum sets.