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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