Mathematics Portfolio

Type II

Creating a Logistic Model

Description

A geometric population model takes the form where r is the growth factor and un is the population at year n. For example, if the population were to increase annually by 20%, the growth factor is r = 1.2, and this would lead to an exponential growth. If r = 1 the population is stable. A logistic model takes a similar form to the geometric, but the growth factor depends on the size of the population and is variable. The growth factor is often estimated as a linear function by taking estimates of the projected initial growth rate and the eventual population.

Part 1

Information which has been given in Part 1: -

- 10,000 fishes are introduced into a lake
- The population increase if 10,000 fishes are introduced into the lake would be by 50% in the first year.
- The long term sustainable limit in this case would be 60,000

It has been given that the geometric population growth model takes the form . Now, if we have to find out the ordered pair (u0,r0): -

- It is mentioned in the description that un is the population at year n. Therefore, u0 will be the population at year 0, or the initial population of the lake which is 10,000.

Therefore, u0 = 10,000 ------------------- (1)

- r is the growth factor as mentioned in the description. As the population would increase by 50% in the first year, so r0 = 1 + 1 x

= 1 + 0.5

= 1.5 ------------------- (2)

So the first ordered pair (u0,r0) as shown in (1) and (2) would be (10000,1.5).

Now, if we have to find out the ordered pair (un, rn): -

- It is given in the question that un = 60,000.

Therefore, un = 60,000 ---------------------- (3)

- r is the growth factor as mentioned in the description. As found out in (3) that un = 60,000 which shows that the population has reached its long term sustainable limit where population is stable.

Therefore, rn = 1 ---------------------- (4)

So the second ordered pair (un , rn) as shown in (3) and (4) would be (60000,1)

A general linear equation has the form (y-y1) = m(x-x1)

where y and x are variables, y1 and x1 are the coordinates of a point on the curve and m is the slope of the curve. To find this equation we shall first find the slope ‘m’ : -

The slope of a curve is given by =

That is, the slope =

As we have found two ordered pairs (u0,r0) and (un , rn) where: -

u0 = X1 = 10,000

r0 = Y1 = 1.5

un = X2 = 60,000

rn = Y2 = 1

Substituting these values in the formula for slope we get = =

= -1 x 10-5

Also, we have one pair of coordinates, that is, (u0, r0) = (10000, 1.5) which is equal to (x1, y1). Substituting all these values in the general linear equation, we get : -

(y – 1.5) = -1 x 10-5(x -10000)

y – 1.5 = -1 x 10-5x + 1 x 10-1

y = -1 x 10-5x + 1.6

Here, the variable y can be replaced by rn which represents the growth factor at some year n and the variable x can be replaced by un which represents the population at some year n. Therefore, the equation for the linear growth factor in terms of un will be: -

rn = (-1 x 10-5)un + 1.6

The equation of the linear growth factor in terms of un can be also be found out by entering the ordered pairs (10000,1.5) and (60000,1) in the STAT mode of the GDC Casio CFX-9850GC Plus. The STAT view looks as follows:-

The graph obtained is as follows: -

In the linear relation found, a is the slope m = -1 x 10-5 and b = y intercept = 1.6

Note: - In all these linear relations obtained from the GDC, we shall take a = m or the slope and b = y intercept.

Also, note that y = rn and x = un as the variable y can be replaced by rn which represents the growth factor at some year n and the variable x can be replaced by un which represents the population at some year n as inferred from the ordered pairs also.

Therefore,

rn = (-1 x 10-5)un + 1.6

where;

un = Population of fish in the lake at some year n.

rn = Population growth rate of fish in the lake at the same year n.

Part 2

The logistic function model for un+1 will be a recursive function in terms of un. We can find out this recursive function using the initial description and our observations in Part 1.

As given in the description, the population growth model takes the form un+1 = rn.un . Replacing rn with the equation of the linear growth factor (rn = (-1 x 10-5)un + 1.6) found in Part 1, we get the following function: -

un+1 = [(-1 x 10-5)un + 1.6]. un

= (-1 x 10-5) un2 + 1.6un

Thus, the logistic function can be modeled by the following recursive function: -

un+1 = (-1 x 10-5) un2 + 1.6un

The graph obtained is as follows:-

Part 3

Entering the recursive logistic function found in Part 2 in the RECUR mode of the GDC Casio CFX-9850GC PLUS, we can find the values of the population over the next 20 years. This can be done by setting the range from 0 to 20. Thus, the following table is obtained: -

The following line graph is obtained using the GDC Casio CFX-9850GC Plus, which is as follows: -

Part 4

As speculated by the biologist, if the initial growth rates vary considerably, then the ordered pair (u0, r0) will change accordingly.

- when initial growth rate r0 = 2

The ordered pair (u0, r0) for the first pair when the predicted growth rate ‘r’ for the year is 2 will be (10000, 2).

(un, rn) = (60000, 1) according to the description.

We can get the equation of the linear growth factor by entering these sets of ordered pairs in the STAT mode of the GDC Casio CFX-9850GC PLUS. The STAT mode looks as follows: -

Thus, we obtain a linear graph which can be modeled by the following function: -