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 mathematical demography was largely developed by Alfred J. Lotka in the early 20th century, building on the earlier work of Leonhard Euler.
is the discrete growth rate, ℓ(a) is the fraction of individuals surviving to age a and b(a) is the number of offspring born to an individual of age a during the time step.
The sum is taken over the entire life span of the organism.
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.
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: 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) = Qert.
Plugging this into the integral equation gives: or This can be rewritten in the discrete case by turning the integral into a sum producing letting
be the boundary ages for reproduction or defining the discrete growth rate λ = er we obtain the discrete time equation derived above: where
Let us write the Leslie matrix as: where
are survival to the next age class and per capita fecundity respectively.
where ℓ i is the probability of surviving to age
weighted by the probability of surviving to age
This implies that By the same argument we find that Continuing inductively we conclude that generally Considering the top row, we get Now we may substitute our previous work for the
terms and obtain: First substitute the definition of the per-capita fertility and divide through by the left hand side: Now we note the following simplification.
we note that This sum collapses to: which is the desired result.
From the above analysis we see that the Euler–Lotka equation is in fact the characteristic 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 expression f as a function of r, we can examine its roots.
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 crosses r = 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 let λ = 1 the discrete formula becomes the replacement rate of the population.