Analysis of a singular functional differential equation that arises from stochastic modeling of population growth

Abstract

The singular functional differential equation x(1 − x) A( x) y′( x) + by( h( x)) − by( x) = − bg( x), x in (0, 1), is studied for initial data y = 0 on x ⩽ a, y continuous on ( a, 1) and y(1−) bounded. The singularity at x = 0+ is removable for a certain class of delayed arguments, h( x). The final end point at x = 1− is the most important singularity because it results in a genuine singular boundary value problem. A formal solution is constructed and is shown to be unique and bounded when g( x) is bounded. A singular decomposition transforms the problem into two nonsingular initial value problems. Singular FDEs of this type arise in the study of the persistence of populations undergoing large random fluctuations when modeled by compound Poisson processes superimposed on logistic-type growth.