A logistic model

IB Mathematics HL Type II Portfolio: Creating a logistic model        International School of Helsingborg – Christian Jorgensen

Creating a logistic  model

Christian Jorgensen

IB Diploma  Programme

IB Mathematics HL Portfolio type  2

Candidate number

International School  of Helsingborg, Sweden

1

IB Mathematics HL Type II Portfolio: Creating a logistic model        International School of Helsingborg – Christian Jorgensen

Theory

A logistic model is expressed as:

un+1  = run

{1}

The growth factor r varies according to un. If r=1 then the population is stable.

Solution

1.   A hydroelectric project is expected to create a large lake into which some fish are to be placed. A biologist estimates that if 1x104 fish were introduced into the lake, the population of fish would increase by 50% in the first year, but the long-term sustainability limit would be about 6 x104. From the information above, write two ordered pairs in the form (u0, r0), (u0, r0) where Un=6x104. Hence, determine the slope and equation of the linear growth in terms of Un

As Un =6 ×104 , the population in the lake is stable. Thus from the definition of a logistic model, r must equal to 1 as un approaches the limit. If the growth in population of fish (initially 1 ×104 ) is 50% during

the first year, r must be equal to 1.5. Hence the ordered pairs are:

(1×104 , 1.5) ,  (6 × 104 ,  1)

One can graph the two ordered pairs. It has been requested to find the growth factor in terms of Un, and thus one should graph the population (x-axis) versus the growth factor (y-axis):

2,5

2,4

2,3

2,2

2,1

2

1,9

1,8

1,7

1,6

1,5

1,4

1,3

1,2

1,1

1

Plot of population U n versus the growth factor r

Trendline

y = -1E-05x + 1,6

0        10000        20000        30000        40000        50000        60000        70000

Fish population Un

Figure 1.1. Graphical plot of the fish population Un versus the growth factor r of the logistic model un+1  = run

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IB Mathematics HL Type II Portfolio: Creating a logistic model        International School of Helsingborg – Christian Jorgensen

The slope of the trend line in figure 1 is determined graphically. The equation of the trend line is of the

form y=mx + b:

y = 1× 105 x + 1.6 . Thus m= 1× 105 . This can be verified algebraically:

m = Δy =

1.0  1.5

= 1× 105

Δx        6 × 104   1× 104         _

The linear growth factor is said to depend upon un and thus a linear equation can be written:

rn  = mun  + c

= (1×105 )(u

) + c

1.5 = (1×105 )(1×104 ) + c

c = 1.5  (1× 105 )(1× 104 ) = 1.6

Hence the equation of the linear growth factor is:

r  = 1× 105 u

+ 1.6

{2}

n        n

2.   Find the logistic function model for un+1

Using equations {1} and {2}, one can find the equation for un+1:

un+1  = run

r  = 1× 105 u

+ 1.6

n        n

un+1

= (1× 105 u

n  + 1.6)un

= (1×105 )(u

)(u

) + 1.6u

The logistic function model for un+1 is:

n        n        n

= (1×105 )(u 2 ) + 1.6u

5        2

un+1  = (1× 10   )(un

) + 1.6un

{3}

3.   Using the model, determine the fish population over the next 20 years and show these values using a line graph

One can determine the population of the first 20 years just by knowing that u1=1×104

u2=1.5×104 . For instance u3 is

and

u  = (1× 105 )(u 2 ) + 1.6u

3        2        2

= (1×105 )(1.5 × 104 )2  + 1.6(1.5 ×104 )

= 2.18 ×104

Table 3.1. The population of fish in a lake over a time range of 20 years estimated using the logistic function model {3}. The interval of calculation is 1 year.

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IB Mathematics HL Type II Portfolio: Creating a logistic model        International School of Helsingborg – Christian Jorgensen

65000

60000

55000

50000

45000

40000

35000

30000

25000

20000

15000

10000

5000

0

Estimated magnitude of population of fish of a hydrolectric project during the first 20 years by means of the logistic function model U n+1

0        1        2        3        4        5        6        7        8        9        10        11        12        13        14        15        16        17        18        19        20

Year

Figure 3.1. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years using logistic function model {3}. The graph is asymptotic approaching 6.0x104 fish despite the years 19 and 20 showing data rounded off to 6.0x104.

4.   The biologist speculates that the initial growth rate may vary considerably. Following the process above, find new logistic function models for un+1 using the initial growth rates r=2, 2.3 and 2.5. Describe any new developments.

a.   For r=2. One can write two ordered pairs (1×104 ,  2.0) ,  (6 × 104 , 1) . The graph of the two ordered pairs is:

4

IB Mathematics HL Type II Portfolio: Creating a logistic model        International School of Helsingborg – Christian Jorgensen

Plot of population U n versus the growth factor r

2,5

2,4

2,3

2,2

2,1

2

1,9

1,8

1,7

1,6

1,5

1,4

1,3

1,2

1,1

1

y = -2E-05x + 2,2

0         10000         20000         30000         40000         50000         60000         70000

Fish population Un

Figure 4.1. Graphical plot of the fish population Un versus the growth factor r of the logistic model un+1 = run

The equation of the trend line is of the form y=mx + b:

can be verified algebraically:

y = 2 × 105 x + 2.2 . Thus m=

2 × 105 . This

m = Δy = Δx

1.0  2.0

6 × 104   1× 104

= 2 × 105

The linear growth factor is said to depend upon un and thus a linear equation can be written:

rn  = mun  + c

= (2 ×105 )(u

) + c

2.0 = (2 ×105 )(1×104 ) + c

c = 2.0  (2 × 105 )(1× 104 ) = 2.2

Hence the equation of the linear growth factor is:

r  = 2 × 105 u

+ 2.2

{4}

n        n

Using equations {1} and {2}, one can find the equation for un+1:

un+1  = run

r  = 2 × 105 u

+ 2.2

n        n

un+1

= (2 ×105 u

n  + 2.2)un

= (2 ×105 )(u

)(u

) + 2.2u

The logistic function model for un+1 is:

n        n        n

= (2 ×105 )(u 2 ) + 2.2u

5        2

un+1  = (2 ×10   )(un

) + 2.2un

{5}

b.   For r=2.3. One can write two ordered pairs (1×104 ,  2.3) ,  (6 ×104 , 1) .  The graph of the two ordered pairs is:

5

IB Mathematics HL Type II Portfolio: Creating a logistic model        International School of Helsingborg – Christian Jorgensen

Plot of population U n versus the growth factor r

2,5

2,4

2,3

2,2

2,1

2

1,9

1,8

1,7

1,6

1,5

1,4

1,3

1,2

1,1

1

y = -3E-05x + 2,56

0         10000         20000         30000         40000         50000         60000         70000

Fish population Un

Figure 4.2. Graphical plot of the fish population Un versus the growth factor r of the logistic model un+1 = run

The equation of the trend line is of the form y=mx + b:

This can be verified algebraically:

y = 3 ×105 x + 2.56 . Thus m=

2 × 105 .

m = Δy = Δx

1.0  2.3

6 × 104   1× 104

= 2.6 ×105

The difference surges because the program used to graph rounds m up to one significant figure.

The linear growth factor is said to depend upon un and thus a linear equation can be written:

rn  = mun  + c

= (2.6 × 105 )(u

) + c

2.3 = (2.6 ×105 )(1×104 ) + c

c = 2.3  (2.6 ×105 )(1× 104 ) = 2.56

Hence the equation of the linear growth factor is:

r  = 2.6 ×105 u

+ 2.56

{6}

n        n

Using equations {1} and {2}, one can find the equation for un+1:

un+1  = run

r  = 2.6 ×105 u

+ 2.56

n        n

u        = (2.6 ×105 u

+ 2.56)u

n+1        n        n

= (2.6 × 105 )(u

)(u

) + 2.56u

The logistic function model for un+1 is:

n        n        n

= (2.6 × 105 )(u 2 ) + 2.56u

5        2

un+1  = (2.6 ×10   )(un

) + 2.56un

{7}

c.   For r=2.5. One can write two ordered pairs (1×104 ,  2.5) ,  (6 × 104 , 1) . The graph of the two ordered pairs is:

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IB Mathematics HL Type II Portfolio: Creating a logistic model        International School of Helsingborg – Christian Jorgensen

Plot of population U n versus the growth factor r

2,5

2,4

2,3

2,2

2,1

2

1,9

1,8

1,7

1,6

1,5

1,4

1,3

1,2

1,1

1

y = -3E-05x + 2,8

0         10000         20000         30000         40000         50000         60000         70000

Fish population Un

Figure 4.3. Graphical plot of the fish population Un versus the growth factor r of the logistic model un+1 = run

The equation of the trend line is of the form y=mx + b:

can be verified algebraically:

y = 3 ×105 x + 2.8 . Thus m=

3 ×105 . This

m = Δy = Δx

1.0  2.5

6 × 104   1× 104

= 3.0 ×105

The linear growth factor is said to depend upon un and thus a linear equation can be written:

rn  = mun  + c

= (3.0 × 105 )(u

) + c

2.5 = (3.0 ×105 )(1×104 ) + c

c = 2.5  (3.0 × 105 )(1× 104 ) = 2.8

Hence the equation of the linear growth factor is:

r  = 3.0 ×105 u

+ 2.8

{8}

n        n

Using equations {1} and {2}, one can find the equation for un+1:

un+1  = run

r  = 3.0 ×105 u

+ 2.8

n        n

u        = (3.0 ×105 u

+ 2.8)u

n+1        n        n

= (3.0 × 105 )(u

)(u

) + 2.8u

The logistic function model for un+1 is:

n        n        n

= (3.0 × 105 )(u 2 ) + 2.8u

5        2

un+1  = (3.0 ×10   )(un

) + 2.8un

{9}

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