File last modified 5 November 1998
Frequently the "type" of box model comes down to the type of equations (differential equations) that describe the system you are modeling. The solution you come up with can vary from analytical to complicated numerical solutions. Because biological models are important in any understanding of the biogeochemical processes that occur at the earth's surface and because they provide good examples of the various types of equations that occur in 0-D box models, we will use them in this section to illustrate some basic "types".
Simple homogeneous ordinary differential equations of one dependent and one independent variable (usually time, the zero-D refers normally to spatial D's). Consider simple birth-death biological models, the deterministic birth model can be written as:
where N is the number of individuals (population) and 
is the birth (growth) rate. A
deterministic death model can be written as:
where 
is the death rate. Simple birth-death models are just the sum of the two:
which, of course, can be solved analytically as
Box models based on analytical solutions like this assume that nothing limits growth, if death exceeds growth then the population will become extinct.
In order to introduce something to limit growth (other than death) one can turn to inhomogeneous equations wherein something in the environment keeps the population from growing without bounds, this can be represented as a "carrying capacity" of the environment. Rewrite the birth-death equation becomes:
where
here r and s are positive constants. This is the simplest linear representation of an inhibitory effect, df/dt must be negative. Recombining we get:
which is also known as the logistic equation. If we let r be the net growth rate in unlimited conditions (absence of regulation) and s be the effect of a N-sized population on growth, then we can define k as the carrying capacity. This gives:
and
Now this has, up to now, all been for a single population. How do the equations change for two (or more) things that interact? Again population dynamics provide some good examples.
Competition comes into play when there are two or more populations competing for the same resource(s). First let's consider the logistic growth equation rewritten for two populations:
where s11 refers to within species competition and s21 between species competition. This can be simplified by assuming that within species and between species competition is basically the same for any given resource (s11 s22=s12 s21).
where p is an amplification factor. Now all of this assumed that the two competing populations were in competition for a resource other than one another. What do the equations look like when a predator-prey type of relation exists? Predation, after all, is only an intense form of competition.
Let's let N1 be the prey population and N2 be the predator population. Then
where r1 is the prey growth rate, r2 the predator death rate, b1 the rate prey is eaten by predators, and b2 the rate at which the predator is successful. This is known as the Lotka-Volterra predator-prey model.
As was stated back in section 13.3.2, any higher order ODE can be rewritten as a set of coupled first order ODEs. Take for example:
if we let dy1/dt=y2 then we have through substitution
as a system of coupled, first order ODEs. This is known as the van der Pol equation.
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