Stable positive periodic solutions of the time-dependent logistic equation under possible hereditary influences

The logistic equation, generalized to include time-dependent but periodic coefficients and a functional, hereditary interaction term, is shown to have a positive periodic solution provided the time-dependent net birth rate has a positive average. Under more restrictive conditions on the interaction term and the net birth rate, this solution is shown to be uniformly asymptotically stable. The approach is to treat the problem as one of the bifurcation of nontrivial positive solutions from the identically zero solution using, roughly speaking, the average of the net birth rate as a nonlinear eigenvalue.