Electron. J. Diff. Eqns., Vol. 2000(2000), No. 02, pp. 1-8.
### Dynamics of logistic equations with non-autonomous bounded coefficients

M. N. Nkashama

**Abstract:**

We prove that the Verhulst logistic equation with
positive non-autonomous bounded coefficients has exactly one bounded
solution that is positive, and that does not approach the zero-solution in
the past and in the future. We also show that this solution is
an attractor for all positive solutions, some
of which are shown to blow-up in finite time backward. Since the
zero-solution is shown to be a repeller for all solutions that remain
below the afore-mentioned one, we obtain an attractor-repeller
pair, and hence (connecting) heteroclinic orbits. The almost-periodic
attractor case is also discussed. Our techniques apply to the critical
threshold-level equation as well.
Submitted October 21, 1999. Published January 1, 2000.

Math Subject Classifications: 34C11, 34C27, 34C35, 34C37, 58F12, 92D25.

Key Words: Non-autonomous logistic equation, threshold-level equation,
positive and bounded solutions, comparison techniques, $\omega$-limit points,
maximal and minimal bounded solutions, almost-periodic functions, separated
solutions.

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M. N. Nkashama

Department of Mathematics, University of Alabama at
Birmingham

Birmingham, Alabama 35294-1170, USA

e-mail: nkashama@math.uab.edu

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