Editing Heinz von Foerster (section)

===The electron tube laboratory===
In 1949, von Foerster started work at the [[University of Illinois at Urbana–Champaign]] at the electron tube laboratory of the Electrical Engineering Department, where he succeeded [[Joseph Tykociński-Tykociner]]. With his students he developed many innovative devices, including ultra-high-frequency electronics<ref>See for example, in ''Review of Scientific Instruments'' '''25''': 640–653, 1954.</ref>

He also worked on mathematical models of population dynamics and in 1959 published a model now called the "[[von Foerster equation]]", which is derivable from the principles of constant aging and conservation of mass.

:<math>\frac{\partial n}{\partial t} + \frac{\partial n}{\partial a}  = - m(a)n, </math>

where: ''n''&nbsp;=&nbsp;''n''(''t'',''a''), ''t'' stands for time and ''a'' for age. ''m''(''a'') is the death in function of the population age; ''n''(''t'',''a'') is the population density in function of age.

When ''m''(''a'')&nbsp;=&nbsp;0, we have:<ref name="ref1">Murray, J.D. ''Mathematical Biology: An Introduction''. Third edition. Interdisciplinary Applied Mathematics. Mathematical Biology. Spring: 2002.</ref>

:<math>\frac{\partial n}{\partial t} = - \frac{\partial n}{\partial a} </math>

It relates that a population ages, and that fact is the only one that influences change in population density.<ref>"Some Remarks on Changing Populations" in ''The Kinetics of Cellular Proliferation'', F. Stohlman, Jr., ed., Grune & Stratton, New York, pp. 382–407 (1959); E. Trucco, ''Bulletin of Mathematical Biophysics'' '''27''': 285–304 and 449–471, 1965</ref>

It is therefore a [[continuity equation]]; it can be solved using the [[method of characteristics]].<ref name="ref1" /> Another way is by [[similarity solution]]; and a third is a numerical approach such as [[finite differences]].

The gross birth rate is given by the following boundary condition:

:<math> n(t,0)= \int_0^\infty b (a)n(t,a) \, dt ,</math>

The solution is only unique given the initial conditions

:<math> n(0,a)= f(a), \, </math>

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