Von Foerster equation

The mathematical formula can be derived from first principles. It reads:

where the population density ${\displaystyle n(t,a)}$ is a function of age ${\displaystyle a}$ and time ${\displaystyle t}$, and ${\displaystyle m(a)}$ is the death function. When ${\displaystyle m(a)=0}$, we have:[2]

${\displaystyle {\frac {\partial n}{\partial t}}=-{\frac {\partial n}{\partial a}}}$

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 ${\displaystyle dt}$ and change in age ${\displaystyle da}$, the population density is:

$${\displaystyle n(t+dt,a+da)=[1-m(a)dt]n(t,a)}$$

That is, during a time period ${\displaystyle dt}$ the population density decreases by a percentage ${\displaystyle m(a)dt}$. Taking a Taylor series expansion to order ${\displaystyle dt}$ gives us that:

$${\displaystyle n(t+dt,a+da)\approx n(t,a)+{\partial n \over {\partial t}}dt+{\partial n \over {\partial a}}da}$$

We know that ${\textstyle da/dt=1}$, since the change of age with time is 1. Therefore, after collecting terms, we must have that:

$${\displaystyle {\partial n \over {\partial t}}+{\partial n \over {\partial a}}=-m(a)n}$$

The von Foerster equation is a continuity equation; it can be solved using the method of characteristics.[2] Another way is by similarity solution; and a third is a numerical approach such as finite differences.

To get the solution, the following boundary conditions should be added:

${\displaystyle n(t,0)=\int _{0}^{\infty }b(a)n(t,a)\,da}$

which states that the initial births should be conserved (see Sharpe–Lotka–McKendrick’s equation for otherwise), and that:

${\displaystyle n(0,a)=f(a)}$

which states that the initial population must be given; then it will evolve according to the partial differential equation.

In Sebastian Aniţa, Viorel Arnăutu, Vincenzo Capasso. An Introduction to Optimal Control Problems in Life Sciences and Economics (Birkhäuser. 2011), this equation appears as a special case of the Sharpe–Lotka–McKendrick’s equation; in the latter there is inflow, and the math is based on directional derivative. The McKendrick’s equation appears extensively in the context of cell biology as a good approach to model the eukaryotic cell cycle.[4]

• Finite difference method
• Partial differential equation
• Renewal theory
• Continuity equation
• Volterra integral equation