Delay Logistic Equation

For , solutions are monotonic. For , the solutions are oscillatory and asymptotically approach . For , the solutions approach a limit cycle. The boundaries can be determined by considering the test solution , which gives the equation ; that has the solution , where is the ProductLog function.

Reference: K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Boston: Kluwer, 1992.