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