DYNAMICS OF LOGISTIC EQUATIONS WITH NON-AUTONOMOUS BOUNDED COEFFICIENTS

We prove that the Verhulst logistic equation with positive non- autonomous bounded coecients has exactly one bounded solution that is positive, and that does not approach the zero-solution in the past and in the future. Wealsoshowthatthissolutionisanattractor forallpositivesolutions, some of which are shown to blow-up innite time backward. Since the zero- solution is shown to be arepeller for all solutions that remainbelow 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.