The mathematical formula can be derived from first principles. It reads:{{Equation box 1|cellpadding|border|indent=:|equation= \frac{\partial n}{\partial t} + \frac{\partial n}{\partial a} = - m(a)n |border colour=#0073CF|background colour=#F5FFFA}}where the population density n(t,a) is a function of age
a and time
t, and m(a) is the death function. When m(a) = 0, we have: :\frac{\partial n}{\partial t} = - \frac{\partial n}{\partial a} It relates that a population ages, and that fact is the only one that influences change in population density; the negative sign shows that time flows in just one direction, that there is no birth and the population is going to die out.
Derivation Suppose that for a change in time dt and change in age da, the population density is:n(t+dt,a + da) = [1-m(a)dt]n(t,a)That is, during a time period dt the population density decreases by a percentage m(a)dt. Taking a
Taylor series expansion to order dt gives us that:n(t+dt,a + da) \approx n(t,a) + {\partial n\over{\partial t}}dt + {\partial n\over{\partial a}}daWe know that da/dt = 1, since the change of age with time is 1. Therefore, after collecting terms, we must have that:{\partial n\over{\partial t}} + {\partial n\over{\partial a}} = -m(a)n ==Analytical solution==