Geodesic
Dynamics of logistic equations with non-autonomous bounded coefficients
Electronic journal of differential equations, Tome 2000 (2000)
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.
Classification : 34C11, 34C27, 34C35, 34C37, 58F12, 92D25
Keywords: 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 
@article{EJDE_2000__2000__a196,
     author = {Nkashama,  M.N.},
     title = {Dynamics of logistic equations with non-autonomous bounded coefficients},
     journal = {Electronic journal of differential equations},
     year = {2000},
     volume = {2000},
     zbl = {0954.34028},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a196/}
}
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%A Nkashama,  M.N.
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%J Electronic journal of differential equations
%D 2000
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Nkashama,  M.N. Dynamics of logistic equations with non-autonomous bounded coefficients. Electronic journal of differential equations, Tome 2000 (2000). http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a196/