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Higher Order Model

Module by: C. Sidney Burrus. E-mail the author

Higher-Order Models

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 higher-order 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

d p d t = r p d p d t = r p
(1)

assumes rr to be the difference between the birth rate and death rate, and that these rates are not a function of time or population.

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 p1p1 be the population of people between zero and ten years of age, p2p2 the population of those from eleven to twenty, p3p3 those from twenty-one to thirty, etc. Let b1b1 be the average birth rate of p1p1, and b2b2 the rate for p2p2, etc., with d1d1 being the average death rate of p1p1, etc. Assume the maximum possible age to be one hundred. The equations for this model are given by

p 1 ( n + 1 ) = b 1 p 1 ( n ) + b 2 p 2 ( n ) + . . . b 10 p 10 ( n ) p ( n + 1 ) = ( 1 - d 1 ) p 1 ( n ) p 2 ( n + 1 ) = ( 1 - d 2 ) p 2 ( n ) p 10 ( n + 1 ) = ( 1 - d 9 ) p 9 ( n ) , ( 66 ) p 1 ( n + 1 ) = b 1 p 1 ( n ) + b 2 p 2 ( n ) + . . . b 10 p 10 ( n ) p ( n + 1 ) = ( 1 - d 1 ) p 1 ( n ) p 2 ( n + 1 ) = ( 1 - d 2 ) p 2 ( n ) p 10 ( n + 1 ) = ( 1 - d 9 ) p 9 ( n ) , ( 66 )
(2)

where the time interval represented by each successive value of nn is the same as that for the age span for each population group, i.e., ten years. Likewise, the birth and death rate are numbers per ten-year period. The equations in Equation 2 can be described by a flow graph, illustrated below for only three sections.

Figure 1
Figure 1 (troff28.png)

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.

p 1 ( n + 1 ) p 2 ( n + 1 ) p 3 ( n + 1 ) p 4 ( n + 1 ) 3 d o t p 10 ( n + 1 ) = b 1 ( 1 - d 1 ) 0 0 3 d o t 0 b 2 0 ( 1 - d 2 ) 0 0 . . . . . . . . . b 9 0 a b o v e ( 1 - d 9 ) b 10 0 3 d o t 0 p 1 ( n ) p 2 ( n ) p 3 ( n ) 3 d o t p 10 ( n ) p 1 ( n + 1 ) p 2 ( n + 1 ) p 3 ( n + 1 ) p 4 ( n + 1 ) 3 d o t p 10 ( n + 1 ) = b 1 ( 1 - d 1 ) 0 0 3 d o t 0 b 2 0 ( 1 - d 2 ) 0 0 . . . . . . . . . b 9 0 a b o v e ( 1 - d 9 ) b 10 0 3 d o t 0 p 1 ( n ) p 2 ( n ) p 3 ( n ) 3 d o t p 10 ( n )
(3)

In compact vector notation, this becomes

P ( n + 1 ) = A P ̲_ ( n ) P ( n + 1 ) = A P ̲_ ( n )
(4)

>From this expression, it is easily seen that the population distribution after nn times ten years from some initial population distribution Punder(0)Punder(0) is given by

P ( n ) = A n P ̲_ ( 0 ) P ( n ) = A n P ̲_ ( 0 )
(5)

There are several interesting observations for the readers with a knowledge of matrix theory. After several steps of nn, the age distribution will assume a form given by the eigenvector of the largest eigenvalue of AA.

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 TT in Euler's method than the span of ages in a population group, and to allow different spans for different groups. Let TT be the time interval between each calculation of new levels. Let SiSi be the span of ages for the group pipi. Consider the population group p2p2. During one time interval TT, the number that dies is d2p2d2p2, the fraction that advances to the next group is TS2TS2 times those that do not die 1-d2p21-d2p2, and the rest 1-TS21-d2p21-TS21-d2p2 stay in the group. The equations become

P 1 = ( n + 1 ) = b 1 p 1 ( n ) + b 2 p 2 ( n ) + ... P 1 =(n+1)= b 1 p 1 (n)+ b 2 p 2 (n)+...
(6)
P 2 ( n + 1 ) = ( T S 1 ) ( 1 - d 1 ) P 1 ( n ) + ( 1 - d 2 ) P 2 ( n ) P 2 (n+1)=( T S 1 )(1- d 1 ) P 1 (n)+(1- d 2 ) P 2 (n)
(7)
P M ( n + 1 ) = ( T S M - 1 ) ( 1 - d M - 1 ) P M - 1 ( n ) + ( 1 - T S M ) ( 1 - d M ) P M ( n ) P M (n+1)=( T S M - 1 )(1- d M - 1 ) P M - 1 (n)+( 1 - T S M )(1- d M ) P M (n)
(8)

The birth rates bibi are the number of births per initial value of population in a group pipi per time interval TT. The death rate is likewise per time interval TT.

This is a rather general formulation that allows non-equal 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 fifth-order 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:

Table 1
N N population
C C capital
A A agriculture
P P pollution
R R non-renewable resources

The model then assumed the form

N ˙ = f 1 ( N , C , A , P , R ) C ˙ = f 2 ( N , C , A , P , R ) A ˙ = f 3 ( N , C , A , P , R ) P ˙ = f 4 ( N , C , A , P , R ) R ˙ = f 5 ( N , C , A , P , R ) N ˙ = f 1 ( N , C , A , P , R ) C ˙ = f 2 ( N , C , A , P , R ) A ˙ = f 3 ( N , C , A , P , R ) P ˙ = f 4 ( N , C , A , P , R ) R ˙ = f 5 ( N , C , A , P , R )
(9)

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 f1f1, f2f2, etc. are developed in a complicated way using the theories and empirical results of specialists in those areas. This generally means that there are numerous intermediate variables defined and used, both for insight and because the data occurs that way. One should not confuse these with the state variables, however. Forrester also uses tables rather than functions to implement the fifi in the simulation. These are usually easier to handle by non-mathematicians, and again in the form that empirical relations are often known.

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].

References

  1. J.W. Forrester. (1971). World Dynamics. Cambridge: Wright-Allen Press.
  2. D.L. Meadows, et al. (1972). The Limits to Growth. New York: Universe Books.
  3. H.S.D. Cole, et. al. (1973). Models of Doom- A Critique of the Limits to Growth. New York: Universe Books.
  4. T.R. Malthus, L.F. Richardson. (1956). Mathematics of Population and Food, and Mathematics of War and Foreign Politics. [edited by J.R. Newman, and Simon and Schuster]. The World of Mathematics, 2,

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