Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics

A local two parameter bifurcation theorem concerning the bifurcation from steady states of time periodic solutions of a nonlinear system of partial, integro-differential equations is proved. A Hopf bifurcation theorem is derived as a corollary. By means of independent and dependent variable changes this theorem is applicable to the general McKendrick equations governing the growth of an age-structured population (with the added feature here of a possible gestation period). The theorem is based on a Fredholm theory developed in the paper for the associated linear equations. An application is given to an age-structured population whose fecundity is density and age dependent and it is shown that for a sufficiently narrow age-specific "reproductive and resource consumption window" steady state instabilities, accompanied by sustained time periodic oscillations, occur when the birth modulus surpasses a critical value.