1.6 An application: age-structured populations

Let \(u(a, t)\) be the density of individuals of age \(a\) at time \(t\), with death rate \(\mu (a)\) and births \(B(t)\) at age \(0\). Individuals age at rate \(1\), so in a short time \(h\) those of age \(a\) at time \(t\) become those of age \(a + h\) at time \(t + h\), depleted by deaths: \(u(a + h, t + h) = u(a, t)(1 - \mu (a)h) + o(h)\). Dividing by \(h\) gives the McKendrick–von Foerster equation \[ u_t + u_a = -\mu (a)u, \qquad u(0, t) = B(t), \qquad u(a, 0) = u_0(a) . \] The characteristics are the lines \(a - t = \text {const}\), the life-lines of cohorts, and along each, \(du/ds = -\mu u\). Writing \(M(a) = \int _0^a\mu \), the solution is \[ u(a, t) = \begin {cases} u_0(a - t)\,e^{-(M(a) - M(a - t))}, & a \geq t \quad \text {(alive at time } 0),\\ B(t - a)\,e^{-M(a)}, & a < t \quad \text {(born after time } 0). \end {cases} \] The two families of characteristics come from the two parts of the boundary of the quadrant \(a, t \geq 0\): the initial line \(t = 0\) and the birth line \(a = 0\). With constant births \(B\), after a time longer than the oldest age the density is the stable age distribution \(Be^{-M(a)}\), the survival curve times the birth rate; this is the life table of demography. When births depend on the population, \(B(t) = \int _0^\infty \beta (a)u(a, t)\,da\) with fertility \(\beta \), the problem becomes a renewal equation, and the population grows or declines exponentially at a rate fixed by Lotka’s equation.

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