Once it becomes necessary to include more than two state variables in a model, and if the interactions are nonlinear, the analytical and phase plane techniques can no longer be used. This section will consider several higherorder models and use digital simulation as the tool for analysis. The first example will be a rather logical extension of some of our earlier population models.
 A. Population Models with Age Specific Birth and Death Rates
Even cursory examination of the assumptions behind the population model assumed in Section IV show them to be unrealistic. The model
assumes
An improvement on this model would allow different birth and death rates to be assigned to members of the population of different ages. This means that the population will have to be divided into groups with similar rates, and that the number of groups necessary will be the number of state variables required. (Reference)
For example, let
where the time interval represented by each successive value of
These equations can be easily programmed and solved on a computer, but because they are linear, there are some interesting properties that can be worked out analytically. They are best seen by writing (Reference) as a matrix equation.
In compact vector notation, this becomes
>From this expression, it is easily seen that the population
distribution after
There are several interesting observations for the readers with a
knowledge of matrix theory.
After several steps of
After several steps, the age distribution will stop changing and this eigenvector is called the stable age distribution, and this largest eigenvalue is the stable growth rate if the eigenvalue is greater than one (decay rate if is is less than one). The problem is a bit more complicated if the eigenvalues are complex (where oscillations occur).
It is possible to modify the equations of Equation 2 to allow for shorter
The birth rates
This is a rather general formulation that allows nonequal age grouping and short time interval without requiring a high order. The system can be posed in matrix form as before. The main limitation on this approach is that it is linear. In general, the various birth and death rates will depend on crowding and other environmental and social factors that are assumed constant here. Even so, insight can be gained into population growth by experiments on these simple linear models.
 B. A Model of the World
One of the most interesting and controversial applications of dynamic modeling is the work of J. Forrester at Massachusetts Institute of Technology on a simulation of the world. In 1970 at the request of an international group called the Club of Rome, Forrester developed a fifthorder model of the work using what he calls "system dynamics," methods that had previously only been applied to industrial and urban systems. The preliminary results were published [1] in 1971, and further work done by his colleague Dennis Meadows was published [2] in 1972. The response to this work was incredible. There has been a flood of articles in newspapers, popular magazines, and scholarly journals – some in praise and others in condemnation. Most have been superficial and emotional. There is, however, one interesting serious response published by a group in England [3] in 1973.
The state variables chosen by Forrester are:

population 

capital 

agriculture 

pollution 

nonrenewable resources 
The model then assumed the form
In one sense, this work is a logical extension of the dynamic modeling discussed in the earlier sections of this paper, and Forrester's formalism is nothing more than using Euler's method to solve simultaneous differential equations. In another sense, his bold use of these methods represents a distinct departure from the specialized models that the demographer, economist, etc. have used in their separate disciplines.
There are several features of Forrester's approach that should be
understood. The functions
A version of the world model has been programmed in APL on an IBM 370 at Rice. The details of this program and instructions on its use are included in the appendix. Examples of the results of the model are given in [1] and [2] and a criticism in [3].