In the study of age-structured population growth, probably one of the most important equations is theEuler–Lotka equation. Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limited in the ability to reproduce), this equation allows for an estimation of how a population is growing.

The field of mathematicaldemography was largely developed byAlfred J. Lotka in the early 20th century, building on the earlier work ofLeonhard Euler. The Euler–Lotka equation, derived and discussed below, is often attributed to either of its origins: Euler, who derived a special form in 1760, or Lotka, who derived a more general continuous version. The equation in discrete time is given by

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

where${\displaystyle \lambda }$ is the discrete growth rate,ℓ(a) is the fraction of individuals surviving to agea andb(a) is the number of offspring born to an individual of agea during the time step. The sum is taken over the entire life span of the organism.

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

LetB(t)dt be the number of births during the time interval fromt tot+dt. Also define thesurvival functionℓ(a), the fraction of individuals surviving to agea. Finally defineb(a) to be the birth rate for mothers of age a. The productB(t-a)ℓ(a) therefore denotes thenumber density of individuals born att-a and still alive att, whileB(t-a)ℓ(a)b(a) denotes the number of births in this cohort, which suggest the followingVolterra 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 timet. We are in effect finding the contributions of all individuals of age up tot. 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 anexponential solution of the formB(t) = Qe^rt. Plugging this into the integral equation gives:

${\displaystyle Q{e}^{rt}={\int }_0^{t}Q{e}^{r(t-a)}\ell (a)b(a)da}$

or

${\displaystyle 1={\int }_0^t{e}^{-ra}\ell (a)b(a)da.}$

This can be rewritten in thediscrete 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 rateλ = 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 sinceb(a) vanishes beyond the boundaries.

Let us write theLeslie matrix as:

where${\displaystyle s_i}$ and${\displaystyle f_i}$ are survival to the next age class and per capita fecundity respectively. Note that${\displaystyle s_i={\ell }_{i+1}/{\ell }_i}$ whereℓ _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 aneigenvalue of thematrix since${\displaystyle {\mathbf{n}}_{\mathbf{i}\mathbf{+}\mathbf{1}}=\mathbf{L}{\mathbf{n}}_{\mathbf{i}}=\lambda {\mathbf{n}}_{\mathbf{i}}}$. 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

${\displaystyle n_{2,t}=\frac{s_1}{\lambda }{n}_{1,t}=\frac{s_0{s}_1}{{\lambda }^2}{n}_{0,t}.}$

Continuinginductively 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

${\displaystyle s_0\dots s_i=\frac{{\ell }_1}{{\ell }_0}\frac{{\ell }_2}{{\ell }_1}\cdots \frac{{\ell }_{i+1}}{{\ell }_i}={\ell }_{i+1}.}$

This sum collapses to:

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

which is the desired result.

From the above analysis we see that the Euler–Lotka equation is in fact thecharacteristic polynomial of the Leslie matrix. We can analyze its solutions to find information about the eigenvalues of the Leslie matrix (which has implications for the stability of populations).

Considering the continuous expressionf as a function ofr, we can examine its roots. We notice that at negative infinity the function grows to positive infinity and at positive infinity the function approaches 0.

The firstderivative is clearly −af and the second derivative isa²f. This function is then decreasing, concave up and takes on all positive values. It is also continuous by construction so by the intermediate value theorem, it crossesr = 1 exactly once. Therefore, there is exactly one real solution, which is therefore the dominant eigenvalue of the matrix the equilibrium growth rate.

This same derivation applies to the discrete case.

If we letλ = 1 the discrete formula becomes thereplacement rate of the population.

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• Demography
• Leonhard Euler
• Integral equations