Euler–Lotka equation

A.J. Lotka in 1911 developed a continuous model of population dynamics as follows. This model tracks only the females in the population.

Let B(t)dt be the number of births during the time interval from t to t+dt. Also define the survival function ℓ(a), the fraction of individuals surviving to age a. Finally define b(a) to be the birth rate for mothers of age a. The product B(t-a)ℓ(a) therefore denotes the number density of individuals born at t-a and still alive at t, while B(t-a)ℓ(a)b(a) denotes the number of births in this cohort, which suggest the following Volterra integral equation for B:

${\displaystyle B(t)=\int _{0}^{t}B(t-a)\ell (a)b(a)\,da.}$

We integrate over all possible ages to find the total rate of births at time t. We are in effect finding the contributions of all individuals of age up to t. We need not consider individuals born before the start of this analysis since we can just set the base point low enough to incorporate all of them.

Let us then guess an exponential solution of the form B(t) = Qe^rt. Plugging this into the integral equation gives:

${\displaystyle Qe^{rt}=\int _{0}^{t}Qe^{r(t-a)}\ell (a)b(a)\,da}$

or

${\displaystyle 1=\int _{0}^{t}e^{-ra}\ell (a)b(a)\,da.}$

This can be rewritten in the discrete case by turning the integral into a sum producing

${\displaystyle 1=\sum _{a=\alpha }^{\beta }e^{-ra}\ell (a)b(a)}$

letting ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ be the boundary ages for reproduction or defining the discrete growth rate λ = e^r we obtain the discrete time equation derived above:

${\displaystyle 1=\sum _{a=1}^{\omega }\lambda ^{-a}\ell (a)b(a)}$

where ${\displaystyle \omega }$ is the maximum age, we can extend these ages since b(a) vanishes beyond the boundaries.

Let us write the Leslie matrix as:

${\displaystyle {\begin{bmatrix}f_{0}&f_{1}&f_{2}&f_{3}&\ldots &f_{\omega -1}\\s_{0}&0&0&0&\ldots &0\\0&s_{1}&0&0&\ldots &0\\0&0&s_{2}&0&\ldots &0\\0&0&0&\ddots &\ldots &0\\0&0&0&\ldots &s_{\omega -2}&0\end{bmatrix}}}$

where ${\displaystyle s_{i}}$ and ${\displaystyle f_{i}}$ are survival to the next age class and per capita fecundity respectively. Note that ${\displaystyle s_{i}=\ell _{i+1}/\ell _{i}}$ where ℓ _i is the probability of surviving to age ${\displaystyle i}$, and ${\displaystyle f_{i}=s_{i}b_{i+1}}$, the number of births at age ${\displaystyle i+1}$ weighted by the probability of surviving to age ${\displaystyle i+1}$.

Now if we have stable growth the growth of the system is an eigenvalue of the matrix since ${\displaystyle \mathbf {n_{i+1}} =\mathbf {Ln_{i}} =\lambda \mathbf {n_{i}} }$. Therefore, we can use this relationship row by row to derive expressions for ${\displaystyle n_{i}}$ in terms of the values in the matrix and ${\displaystyle \lambda }$.

Introducing notation ${\displaystyle n_{i,t}}$ the population in age class ${\displaystyle i}$ at time ${\displaystyle t}$, we have ${\displaystyle n_{1,t+1}=\lambda n_{1,t}}$. However also ${\displaystyle n_{1,t+1}=s_{0}n_{0,t}}$. This implies that

${\displaystyle n_{1,t}={\frac {s_{0}}{\lambda }}n_{0,t}.\,}$

By the same argument we find that

${\displaystyle n_{2,t}={\frac {s_{1}}{\lambda }}n_{1,t}={\frac {s_{0}s_{1}}{\lambda ^{2}}}n_{0,t}.}$

Continuing inductively we conclude that generally

${\displaystyle n_{i,t}={\frac {s_{0}\cdots s_{i-1}}{\lambda ^{i}}}n_{0,t}.}$

Considering the top row, we get

${\displaystyle n_{0,t+1}=f_{0}n_{0,t}+\cdots +f_{\omega -1}n_{\omega -1,t}=\lambda n_{0,t}.}$

Now we may substitute our previous work for the ${\displaystyle n_{i,t}}$ terms and obtain:

${\displaystyle \lambda n_{0,t}=\left(f_{0}+f_{1}{\frac {s_{0}}{\lambda }}+\cdots +f_{\omega -1}{\frac {s_{0}\cdots s_{\omega -2}}{\lambda ^{\omega -1}}}\right)n_{(0,t)}.}$

First substitute the definition of the per-capita fertility and divide through by the left hand side:

${\displaystyle 1={\frac {s_{0}b_{1}}{\lambda }}+{\frac {s_{0}s_{1}b_{2}}{\lambda ^{2}}}+\cdots +{\frac {s_{0}\cdots s_{\omega -1}b_{\omega }}{\lambda ^{\omega }}}.}$

Now we note the following simplification. Since ${\displaystyle s_{i}=\ell _{i+1}/\ell _{i}}$ we note that

${\displaystyle s_{0}\ldots s_{i}={\frac {\ell _{1}}{\ell _{0}}}{\frac {\ell _{2}}{\ell _{1}}}\cdots {\frac {\ell _{i+1}}{\ell _{i}}}=\ell _{i+1}.}$

This sum collapses to:

${\displaystyle \sum _{i=1}^{\omega }{\frac {\ell _{i}b_{i}}{\lambda ^{i}}}=1,}$

which is the desired result.