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:
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]
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 and change in age , the population density is:
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:
which states that the initial births should be conserved (see Sharpe–Lotka–McKendrick’s equation for otherwise), and that:
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]
See also[]
- Finite difference method
- Partial differential equation
- Renewal theory
- Continuity equation
- Volterra integral equation
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.
- Diffusion
- Parabolic partial differential equations
- Stochastic differential equations
- Transport phenomena
- Equations of physics
- Mathematical and theoretical biology
- Ecology