Leslie Population Models

Ma 2071

 

 

Problem: use linear algebra to predict the growth and characteristics of a population which is broken into age groups.  This will illustrate the numerical use of diagonalization. Conference yesterday illustrated a geometric use of diagonalization.

 

Knowing the total population is good but knowing how it is distributed among different age groups is better. Different groups have different social and economic needs which need planning and resources for:

 

      • young children need daycare
      • older children need schools, playgrounds
      • adults provide taxes, have jobs, buy houses and consumer goods, and purchase and drive cars
      • older adults require health care and different housing
      • industries need utilities, good roads and zoning

 

 

        The general study of these characteristics is called demographics.  It is of concern to civil engineers who must plan public utilities, computer scientists who provide information services, mechanical engineers who design transportation media such as cars and trains, electrical engineers who must provide adequate electricity, biotech people who provide solutions to the unique health needs of each group, and mathematicians (as actuaries) who work with various kinds of insurance needed by all groups (auto, life, house, health).  Electrical and computer engineers also need markets for consumer goods they design such as computer games, Internet access,  high definition television and cell phones. 

 

       In this course, the Bio majors are looking at the spread of HIV and also what it might take to effectively change that.

 

         Quantitative models are of considerable value.  Knowing in advance that the population of 20 to 30 year olds is going to decline by 2% is of value to an auto manufacturer who has a line of sports cars. In Maine, the population of school age children has dropped by 16% in the past 10 years. This information is useful to state and local planners looking at school, teachers, taxes and people to eventually fill jobs.  Demographics.

______________________________________________

The mathematics:    x is a column vector with the size of the population for each group as its components.  In this hypothetical model, the first component of x might be the number of 0 to 25 year olds, the second component 26 to 50 year olds, the third 51 to 75 year olds and the last 76 year old or more.

 

The future populations are generated according to the matrix model

 

                   x(k+1) = A x(k)

 

 A  is called a Leslie Matrix.  It has birthrates of each of the age groups along the top row and has survival percentages along the subdiagonal.

It is 4x4 in this instance. I will use Maple to study this below…

 

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

 

> with(plots):

Intitial Population - divided into 4 age groups

 

 

          (lots of young people)

Warning, the name changecoords has been redefined

 

> x0:=matrix(4,1,[10,2,3,1]);

This can be displayed in histogram form:

 

> matrixplot(x0,heights=histogram,axes=frame,gap=0.25,style=patch);

 

The following matrix is used to compute the population one generation into the future.

> A:=matrix(4,4,[.2,1.5,.7,.2,.9,0,0,0,0,.88,0,0,0,0,.84,0]);

Transient Behavior - what happens in the immediate future:

> x1:=multiply(A,x0);

 

> matrixplot(x1,heights=histogram,axes=frame,gap=0.25,style=patch);

 

 

 

> x2:=multiply(A,x1);  2 units of time into the future

>

 

 

> matrixplot(x2,heights=histogram,axes=frame,gap=0.25,style=patch);

 

 

Now let's look further into the future; 10 units of time.

> x10:=multiply(A^9,x0);

> x11:=multiply(A^10,x0);

 

 

 

Let's compare how each component of the population has changed:

 

> ratio1:=x11[1,1]/x10[1,1];

> ratio2:=x11[2,1]/x10[2,1];

> ratio3:=x11[3,1]/x10[3,1];

> ratio4:=x11[4,1]/x10[4,1];

 

 

Next, we plot both populations on the same histogram:

> P:=augment(x10,x11):

> matrixplot(P,heights=histogram,axes=frame,gap=0.25,style=patch);

 

 

What does this last histogram show?  It shows that the shape remains the same, from one period of time to the next.  All groups increase in size but they increase in size by the same factor (1.44)

What does this have to do with diagonalization?   The vector made up of the sizes of the age groups is an eigenvector of A.  The factor that they increase by is an eigenvalue of A,  1.45. More specifically, it is the largest of the four eigenvalues of A, the dominant eigenvalue.

What the above histogram shows is algebraically shown as

x(11) = 1.45 x(10)

and since                   x(11) = A x(10)

this means  1.45 is an eigenvalue of A  and x(10)  is its eigenvector.

 

Furthermore, this behavior continues indefinitely.  This means  x(21) = 1.45 x(20)  or   x(25) = 1.45 x(24).  In other words, the shape of the histogram remains the same and the population increases by a factor of 1.45 each time (i.e. goes up by 45%)

 

The moral?  If we know the largest eigenvalue and its eigenvector for a dynamical system, then we know what the long term behavior of the system will be.

                   The profile of the solution will be determined by the eigenvector

                   The rate of growth will be determined by the eigenvalue.

 

These remarks apply to any dynamical system of the form   x(k+1) = A x(k).

 

All of this can be quickly predicted by computing the eigenvalues and eigenvectors of A:

> eigenvectors(A);

 

 

 

 

 

Amidst  this barrage of  data is a largest eigenvalue,  1.444463...   and  its eigenvector,  [.79, .49, .3 , .175]   Its components have the same relative size as x(11) and x(10) above (you might check this with a calculator!) 

 

                                                                                                          

To see why it works out this way, we need to look at  the basic equation   A = P D P-1  We will do that Friday…