The authors analyze a linear, age-structured population model with time delay. The model has the form
where p(a,t) denotes the age density distribution at time t and age a, (a) is the age dependent mortality, is the maximum age attained by individuals, k(a) is the female sex ratio at age a, h(a) is the age dependent fertility modulus, is the fertility period of females, is the specific fertility rate of females, and r is the time delay.
A semigroup of operators in the Banach space is associated with this problem and its infinitesimal generator is identified. Spectral information about the infinitesimal generator is used to prove that the solutions have the asymptotic behavior
where is the unique real solution of the characteristic equation
and is a constant. It is proved that when the solutions approach equilibrium in an oscillatory manner. A nonlinear version of the model is also analyzed. The balance equation is given an additional mortality term of the form K f(N(t))p(a,t), where N(t) is the total population at time t and f is a logistic-type function. The authors prove the existence and uniqueness of solutions to the nonlinear problem.