Von Foerster equation

The McKendrick–von Foerster equation is a linear first-order partial differential equation encountered in several areas of mathematical biology – for example, demography and cell proliferation modeling; it is applied when age structure is an important feature in the mathematical model.^[1] It was first presented by Anderson Gray McKendrick in 1926 as a deterministic limit of lattice models applied to epidemiology, and subsequently independently in 1959 by biophysics professor Heinz von Foerster for describing cell cycles.

Mathematical formula[]

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

{\frac {\partial n}{\partial t}}+{\frac {\partial n}{\partial a}}=-m(a)n

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:^[1]

{\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}}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:

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

Analytical solution[]

The von Foerster equation is a continuity equation; it can be solved using the method of characteristics.^[1] 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:

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:

n(0,a)=f(a)

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

Similar equations[]

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 ; 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.^[2]

References[]

^ ^a ^b ^c MURRAY, J.D. Mathematical Biology: an introduction. third edition. Interdisciplinary Applied Mathematics. Mathematical Biology. Spring: 2002.
^ Gavagnin, Enrico (14 October 018). "The invasion speed of cell migration models with realistic cell cycle time distributions". Journal of Theoretical Biology. 79 (1): 91–99. arXiv:1806.03140. doi:10.1016/j.jtbi.2018.09.010. PMID 30219568.

[ref1-1] MURRAY, J.D. Mathematical Biology: an introduction. third edition. Interdisciplinary Applied Mathematics. Mathematical Biology. Spring: 2002.

[2] Gavagnin, Enrico (14 October 018). "The invasion speed of cell migration models with realistic cell cycle time distributions". Journal of Theoretical Biology. 79 (1): 91–99. arXiv:1806.03140. doi:10.1016/j.jtbi.2018.09.010. PMID 30219568.

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