Euler–Lotka equation

Example

Consider an age-structured population, and let $a \ge 0$ denote age. Let $l(a)$ be the probability of surviving to age $a$, and $m(a)$ the birth rate of individuals of age $a$. Assume $l m$ is nonnegative, integrable on $[0, \infty)$ and not almost everywhere zero. The birth rate $B(t)$ at time $t \ge 0$ satisfies a renewal equation $$B(t) = G(t) + \int_0^t B(t - a) \, l(a) \, m(a) \, da,$$ where $G(t)$ collects the births from individuals already alive at time $0$. This is a Volterra equation of convolution type with kernel $k = l m$.

Looking for exponential solutions $B(t) = Q e^{rt}$ of the equation without the term $G$, with the integral taken over all ages $a \ge 0$, gives the Euler–Lotka equation $$\int_0^\infty e^{-r a} \, l(a) \, m(a) \, da = 1.$$ Its left side is the Laplace transform $\hat k(r)$ of the kernel, so $r$ is a root of $1 - \hat k(r) = 0$, the denominator of the Laplace transform method.

Check on an instance: let $l(a) m(a) = b \, e^{-\mu a}$ with constants $b > 0$ and $\mu \ge 0$. For $r > -\mu$ the left side is $\int_0^\infty b \, e^{-(r + \mu) a} \, da = \frac{b}{r + \mu}$. It equals $1$ exactly when $r = b - \mu$, the birth rate minus the death rate.