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A semigroup approach to age-dependent population dynamics with time delay. (English) Zbl 0691.92014

The authors analyze a linear, age-structured population model with time delay. The model has the form

$\partial p\left(a,t\right)/\partial t+\partial p\left(a,t\right)/\partial a=-\mu \left(a\right)p\left(a,t\right),\phantom{\rule{1.em}{0ex}}00,$
$p\left(a,\theta \right)={p}_{0}\left(a,\theta \right),\phantom{\rule{1.em}{0ex}}0\le a\le {a}_{m},\phantom{\rule{1.em}{0ex}}-r\le \theta \le 0,$
$p\left(0,t\right)=\beta {\int }_{{a}_{1}}^{{a}_{2}}k\left(a\right)h\left(a\right)p\left(a,t-r\right)da,\phantom{\rule{1.em}{0ex}}t>0,$

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, $\left[{a}_{1},{a}_{2}\right]$ 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 $C\left(\left[0,{a}_{m}\right]×\left[-r,0\right]\right)$ 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

$p\left(a,t\right)=C\left({p}_{0}\right)exp\left[{\lambda }_{0}\left(t-a\right)-{\int }_{0}^{a}\mu \left(b\right)db\right]+o\left(exp\left[\left({\lambda }_{0}-ϵ\right)t\right]\right),$

where ${\lambda }_{0}$ is the unique real solution of the characteristic equation

$1=\beta {\int }_{{a}_{1}}^{{a}_{2}}k\left(a\right)h\left(a\right)exp\left[-\lambda \left(a+r\right)-{\int }_{0}^{a}\mu \left(b\right)db\right]da$

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.

Reviewer: G.F.Webb

##### 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