*(English)*Zbl 0691.92014

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, $\mu $ (a) is the age dependent mortality, ${a}_{m}$ 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, $[{a}_{1},{a}_{2}]$ is the fertility period of females, $\beta $ is the specific fertility rate of females, and r is the time delay.

A semigroup of operators in the Banach space $C([0,{a}_{m}]\times [-r,0])$ 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 ${\lambda}_{0}$ is the unique real solution of the characteristic equation

and $C\left({p}_{0}\right)$ is a constant. It is proved that when $\lambda =0$ 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.

##### MSC:

92D25 | Population dynamics (general) |

35Q99 | PDE of mathematical physics and other areas |

35B40 | Asymptotic behavior of solutions of PDE |

47D03 | (Semi)groups of linear operators |

47H20 | Semigroups of nonlinear operators |

35P99 | Spectral theory and eigenvalue problems for PD operators |