Lotka's stable population equations (cont.)

Lotka’s third equation: the characteristic equation of a stable population (cont.)

However re-writing the equation making use of the fact that for any number z, exp{-z} = 1 / exp{z} , we can better understand the implications of the relationship between r, f_x and L_x

$$1={\displaystyle \sum }_{15}^{50}\frac{L_x\cdot f_x}{exp\left\{r(x+0.5)\right\}}= \frac{L_{15}\cdot f_{15}}{exp\left\{r15.5\right\}}+\frac{L_{16}\cdot f_{16}}{exp\left\{r16.5\right\}}+\frac{L_{17}\cdot f_{17}}{exp\left\{r17.5\right\}}+\cdots +\frac{L_{49}\cdot f_{49}}{exp\left\{r49.5\right\}}$$

This is the expanded version of Lotka’s third equation, assuming that the reproductive age span runs from ages 15 to 50.

Note that when r is zero, exp{rz} is exactly 1 for any value of z, since exp{0} = 1.  This means that for the special case of a stationary population (r=0):

$$1={\displaystyle \sum }_{15}^{50}{L}_x\cdot f_x=NRR$$

In other words, Lotka’s equation will balance in a stationary population when the growth rate is zero and the NRR = 1.  We have already encountered this relationship in the module on reproductivity .

When we look at the L_x ⋅ f_x pairs in the expanded version of the equation, we can see that an increase in the L_x ⋅ f_x values corresponding to NRR>1 will require increased values of the exp { r (x + 0.5) } terms in the denominators to keep the sum of the terms equal to 1.0.  To increase the value of these exponential terms, r must be positive (r > 0).  Conversely, if the L_x ⋅ f_x values are lower, corresponding to an NRR<1, lower values of the exp { r (x + 0.5) } terms in the denominators will be required to keep the sum of the terms equal to 1.0. 

One way to "solve" the characteristic equation for a stable population is to guess a value of r, calculate the sum of the

$$\frac{L_x\cdot f_x}{exp\left\{r(x+0.5)\right\}}$$

terms. If the sum is larger than 1.0, then increase the value of r to decrease each of these terms, if the sum is smaller than 1.0, then decrease the value of r to increase each of the terms. This is known as an "iterative" solution – each iteration should produce a value of r that is closer to the true value needed to balance the equation.