Mathematics Higher Level

Assignment: Mathematics Portfolio Type II:

Creating a Logistic Model

School: Trinity Grammar School

School Code:

Name of Student: Cassian Ho

Student Number:

Introduction

If a hydroelectric project is expected to cerate a large lake into which some fish are to be placed, a biologist estimates that if 10,000 fish were introduced into the lake, the population of fish would increase by 50% in the first year, but the long-term sustainable limit would be about 60,000.

In order to estimate the growth rate of the population of fish, it is best to find a linear growth factor for. We do this by finding two ordered pairs in the form (u0, r0), (un, rn).

rn is the growth rate when the population is n.

since we know from the information given that when there are 60000 fish in the lake, the growth rate is stable i.e. 1, this can be represented as one of our ordered pairs: (60000, 1). We also know that when 10000 fish are in the lake, the growth rate is 50% i.e. population is multiplied by 1.5, so the second ordered pair is (10000, 1.5).

from these pairs we have:

r10000 = 1.5

r60000 = 1

and since we are trying to search for a linear function of the growth rate, we can also denote rn as:

rn = mn + b where n = population of fish. Substituting the two ordered pairs, we have:

1.5 = m(10000) + b

1 = m(60000) + b

Putting this into the GDC, we find that the solutions to the two unknowns are:

m = -0.00001

b = 1.6

from this, we can make the conjecture for the linear growth factor of fish in terms of Un:

rn = -0.00001 × un + 1.6

Since geometric population growth models take the form: un+1 = r × un

and we also have that rn = -0.00001 × un + 1.6, we find that the function for un+1 is:

un+1 = (-0.00001 × un + 1.6) × un

This is rather obvious because population size of the next year is equal to population size this year multiplied by the growth rate.

Now, let us explore what would happen if we left this population of fish to grow for 20 years, assuming that our models for total population and growth rate are correct. It is best to use Microsoft Excel 2007 to calculate this. Initially there is a population of 10000:

(This data was achieved by using formulas in Microsoft Excel)

With these data values, it is possible to find a logistic function for the population size of fish by using the GDC. Based on the values given in the table above, the calculator is able to estimate the logistic function of un+1. The function given by the calculator is:

However, the function that the calculator gives is just an estimate, so it is safe for us to round off this function to:

y =

Putting the data obtained from the excel table into a line graph, we have:

This graph shows that starting from a population of 10000 fish, the rate of growth will increase initially, and then decrease so that at approximately year 10, there is a stable growth rate. This is an S-shaped curve.

Next we should consider how a different initial growth rate will affect the graph. For example, it may be the case that biologists speculate that the initial growth rate of fish may vary considerably. I will now investigate functions models for un+1 with growth rates r = 2, 2.3, 2.5.

Model for Growth Rate = 2

When we have a new initial growth rate, the ordered pairs i.e. (un, rn) have now changed to:

(60000, 1)

(10000, 2)

To find the linear growth factor, we must form the two equations:

2 = m(10000) + b

1 = m(60000) + b

Solving this on the GDC, we have m = -0.00002 and b = 2.2 so the linear growth factor is:

rn = -0.00002 × un + 2.2

and therefore the function for un+1 for is:

un+1 = (-0.00002 × un + 2.2) un

If we let 10000 fish with an initial growth rate of 2 cultivate for 20 years, the population of growth would be:

As we have done before, we should also use the GDC to find an estimate for the logistic function:

However, the function that the calculator gives is just an estimate, so it is safe for us to round off this function to:

y =

Putting this in a graph, we have:

Model for Growth Rate = 2.3

When we have a new initial growth rate, the ordered pairs i.e. (un, rn) have now changed to:

(60000, 1)

(10000, 2.3)

To find the linear growth factor, we form the two equations:

2.3 = m(10000) + b

1 = m(60000) + b

Solving this on the GDC, we have m = -0.000026 and b = 2.56 so the linear growth factor is:

rn = -0.000026 × un + 2.56

and therefore the function for un+1 for is:

un+1 = (-0.000026 × un + 2.56) un

If we let 10000 fish with an initial growth rate of 2.3 cultivate for 20 years, the population of growth would be:

As we have done before, we should also use the GDC to find an estimate for the logistic function:

However, the function that the calculator gives is just an estimate, so it is safe for us to round off this function to:

y =

Putting this data into a graph, we have:

Model for Growth Rate = 2.5

When we have a new initial growth rate, the ordered pairs i.e. (un, rn) have now changed to:

(60000, 1)

(10000, 2.5)

To find the linear growth factor, we form the two equations:

2.5 = m(10000) + b

1 = m(60000) + b

Solving this on the GDC, we have m = -0.00003 and b = 2.8 so the linear growth factor is:

rn = -0.00003 × un + 2.8

and therefore the function for un+1 for is:

un+1 = (-0.00003 × un + 2.8) un

If we let 10000 fish with an initial growth rate of 2.5 cultivate for 20 years, the population of growth would be:

As we have done before, we should also use the GDC to find an estimate for the logistic function:

y =

Putting these data values into a graph, we have:

For now, let us compare these graphs to make generalizations about how the initial growth rate can effect the population of fish over time.

The next page shows graphs the population of fish starting from initial growth rates r = 1.5, 2, 2.3 and 2.5.

From these graphs, we can make a few generalizations:

Generalization 1: As initial growth rate increases, the initial slope of the curve becomes steeper.