Dynamics of logistic equations with non-autonomous bounded coefficients

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.

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