0.7 Higher order model

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 phaseplane techniques can no longer be used. This section will consider several higher-order models and use digital simulation as the tool foranalysis. 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 $r$ 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 thenumber of state variables required. [link]

For example, let $p_1$ be the population of people between zero and ten years of age, $p_2$ the population of those from eleven to twenty, $p_3$ those from twenty-one to thirty, etc. Let $b_1$ be the average birth rate of $p_1$ , and $b_2$ the rate for $p_2$ , etc., with $d_1$ being the average death rate of $p_1$ , etc. Assume the maximum possible age to be one hundred.The equations for this model are given by

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

These equations can be easily programmed and solved on a computer, but because they are linear, there are some interesting properties that canbe worked out analytically. They are best seen by writing [link] as a matrix equation.

In compact vector notation, this becomes

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

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

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 [link] to allow for shorter $T$ in Euler's method than the span of ages in a population group, and to allow different spans for different groups. Let $T$ be the time interval between each calculation of new levels. Let $S_i$ be the span of ages for the group $p_i$ . Consider the population group $p_2$ . During one time interval $T$ , the number that dies is $d_2{p}_2$ , the fraction that advances to the next group is $\left(,\frac{T}{S_2},\right)$ times those that do not die $\left(,1,,,-,,,d_2,\right)p_2$ , and the rest $\left(,1,,,-,,,\frac{T}{S_2},\right)\left(,1,,,-,,,d_2\right)p_2$ stay in the group. The equations become