mathinvestigationii - logistic model

8/8/2019 MATHinvestigationII - Logistic Model

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

1

Creating a logistic model

Christian Jorgensen

IB Diploma Programme

IB Mathematics HL Portfolio type 2Candidate numbervvvvvvvvvv

International School of Helsingborg, Sweden


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Theory

A logistic model is expressed as:

1n nu ru+ = {1}

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

Solution1. 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 6x104. 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 =6410 , the population in the lake is stable. Thus from the definition of a logistic model, rmust

equal to 1 as un approaches the limit. If the growth in population of fish (initially 1410 ) is 50% during

the first year, rmust be equal to 1.5. Hence the ordered pairs are:4 4

(1 10 , 1.5) , (6 10 , 1)

One can graph the two ordered pairs. It has been requested to find the growth factorin terms of Un,

and thus one should graph the population (x-axis) versus the growth factor (y-axis):

Plot of population Un versus the growth factor r

Trendline

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

11,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

2,12,2

2,3

2,4

2,5

0 10000 20000 30000 40000 50000 60000 70000

Fish population Un

Growthfactorr

Figure 1.1. Graphical plot of the fish population Un versus the growth factorrof the logistic model 1n nu ru+ =


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

formy=mx + b: 51 10 1.6y x= + . Thus m= 51 10 . This can be verified algebraically:

4 4

1.0 1.5

6 10 1 10

ym

= = =

51 10

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

5

5 4

5 4

( 1 10 )( )

1.5 ( 1 10 )(1 10 )

1.5 ( 1 10 )(1 10 ) 1.6

n n

n

r mu c

u c

c

c

= +

= +

= +

= =

Hence the equation of the linear growth factor is:51 10 1.6n nr u

= + {2}

2. Find the logistic function model for un+1

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

5

5

1

5

5 2

1 10 1.6

( 1 10 1.6)

( 1 10 )( )( ) 1.6

( 1 10 )( ) 1.6

n n

n n

n n n

n n n

n n

u ru

r u

u u u

u u u

u u

+

+

=

= +

= +

= +

= +

The logistic function model forun+1 is:5 2

1 ( 1 10 )( ) 1.6n n nu u u

+= + {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=1410 and

u2=1.5410 . For instance u3 is

5 2

3 2 2

5 4 2 4

4

( 1 10 )( ) 1.6

( 1 10 )(1.5 10 ) 1.6(1.5 10 )

2.18 10

u u u

= +

= +

=

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 ofcalculation is 1 year.

Year Population Year Population

1 1.0 410 11 5.98 410 2 1.5 410 12 5.99 410 3 2.18 410 13 5.996 410 4 3.01 410 14 5.999 410 5 3.91 410 15 5.9994 410 6 4.72 410 16 5.9997 410 7 5.34 410 17 5.9999 410


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

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

2,1

2,2

2,3

2,4

2,5

0 10000 20000 30000 40000 50000 60000 70000

Fish population Un

Growthfactorr


The equation of the trend line is of the formy=mx + b: 52 10 2.2y x= + . Thus m= 52 10 . This

can be verified algebraically:

4 4

1.0 2.0

6 10 1 10

ym

= = =

52 10


5

5 4

5 4

( 2 10 )( )

2.0 ( 2 10 )(1 10 )

2.0 ( 2 10 )(1 10 ) 2.2

n n

n

r mu c

u c

c

c

= +

= +

= +

= =

Hence the equation of the linear growth factor is:52 10 2.2n nr u

= + {4}

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

1

5

5

15

5 2

2 10 2.2

( 2 10 2.2)( 2 10 )( )( ) 2.2

( 2 10 )( ) 2.2

n n

n n

n n n

n n n

n n

u ru

r u

u u uu u u

u u

+

+

=

= +

= +

= +

= +


1 ( 2 10 )( ) 2.2n n nu u u

+= + {5}

b. Forr=2.3. One can write two ordered pairs 4 4(1 10 , 2.3) , (6 10 , 1) . The graph of the

two ordered pairs is:


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

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

2,1

2,2

2,3

2,4

2,5

0 10000 20000 30000 40000 50000 60000 70000

Fish population Un

Growthfactorr


The equation of the trend line is of the formy=mx + b: 53 10 2.56y x= + . Thus m= 52 10 .

This can be verified algebraically:

4 4

1.0 2.3

6 10 1 10

ym

= = =

52.6 10

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


5

5 4

5 4

( 2.6 10 )( )

2.3 ( 2.6 10 )(1 10 )

2.3 ( 2.6 10 )(1 10 ) 2.56

n n

n

r mu c

u c

c

c

= +

= +

= +

= =

Hence the equation of the linear growth factor is:52.6 10 2.56n nr u

= + {6}


1

5

5

1

5

5 2

2.6 10 2.56

( 2.6 10 2.56)

( 2.6 10 )( )( ) 2.56

( 2.6 10 )( ) 2.56

n n

n n

n n n

n n n

n n

u ru

r u

u u u

u u u

u u

+

+

=

= +

= +

= +

= +


1 ( 2.6 10 )( ) 2.56n n nu u u

+= + {7}

c. Forr=2.5. One can write two ordered pairs 4 4(1 10 , 2.5) , (6 10 , 1) . The graph of the

two ordered pairs is:


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

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

2,1

2,2

2,3

2,4

2,5

0 10000 20000 30000 40000 50000 60000 70000

Fish population Un

Growthfactorr


The equation of the trend line is of the formy=mx + b: 53 10 2.8y x

= + . Thus m= 53 10 . This

can be verified algebraically:

4 4

1.0 2.5

6 10 1 10

ym

= = =

53.0 10


5

5 4

5 4

( 3.0 10 )( )

2.5 ( 3.0 10 )(1 10 )

2.5 ( 3.0 10 )(1 10 ) 2.8

n n

n

r mu c

u c

c

c

= +

= +

= +

= =


= + {8}


1

5

5

1

5

5 2

3.0 10 2.8

( 3.0 10 2.8)

( 3.0 10 )( )( ) 2.8

( 3.0 10 )( ) 2.8

n n

n n

n n n

n n n

n n

u ru

r u

u u u

u u u

u u

+

+

=

= +

= +

= +

= +


1 ( 3.0 10 )( ) 2.8n n nu u u

+= + {9}


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The findings for a, b and c enables one to calculate the magnitude of the population for three studies with

different growth factor. These results are shown schematically and graphically:

Table 4.1.Growth factorr=2.0 The population of fish in a lake over a time range of 20 years estimated using the logistic function model

{5}. The interval of calculation is 1 year. Please note that from year 6 and onwards the population resonates above and below the limit (yetdue to the rounding up of numbers this is not observed), and it finally stabilizes in year 17 (and onwards) where the population is exactly

equal to the sustainable limit.Year Population Year Population

1 1.0 410 11 6.00 410 2 2.0 410 12 6.00 410 3 3.60 410 13 6.00 410 4 5.33 410 14 6.00 410 5 6.04 410 15 6.00 410 6 5.99 410 16 6.00 410 7 6.00 410 17 6.00 410 8 6.00 410 18 6.00 410 9 6.00 410 19 6.00 410 10 6.00 410 20 6.00 410

Estimated magnitude of population of fish of a hydrolectric project during

the first 20 years by means of the logistic function model Un+1 {5}

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000

65000

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

Year

PopulationU

n+1/fish

Figure 4.1.Growth factorr=2.0. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years

using logistic function model {5}.


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Table 4.2.Growth factorr=2.3 The population of fish in a lake over a time range of 20 years estimated using the logistic function model

{7}. The interval of calculation is 1 year.


1 1.0 410 11 6.00 410 2 2.3 410 12 6.00 410 3 4.51 410 13 6.00 410

4 6.26 410 14 6.00 410 5 5.84 410 15 6.00 410 6 6.08 410 16 6.00 410 7 5.95 410 17 6.00 410 8 6.03 410 18 6.00 410 9 5.98 410 19 6.00 410 10 6.01 410 20 6.00 410



0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000

65000

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

Year

PopulationU

n+1/fish



Table 4.3.Growth factorr=2.5. The population of fish in a lake over a time range of 20 years estimated using the logistic function model{9}. The interval of calculation is 1 year.


1 1.0 410 11 5.90 410 2 2.5 410 12 6.08 410 3 5.13 410 13 5.94 410 4 6.47 410 14 6.05 410 5 5.56 410 15 5.96 410 6 6.30 410 16 6.03 410 7 5.74 410 17 5.97 410


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8 6.19 410 18 6.02 410 9 5.84 410 19 5.98 410 10 6.12 410 20 6.01 410



0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000

65000

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

Year

P

opulationU

n+1/fish



The logistic function models attain greater values form and c when the growth factorrincreases in

magnitude. As a consequence, the initial steepness of graphs 4.1, 4.2 and 4.3 increases, and thus thegrowth of the population increase. Also, as these values increase the population (according to this

model) goes above the sustainable limit but then drops below the subsequent year so as to stabilize.There is no longer an asymptotic relationship (going from {5} to {9}) but instead a stabilizing

relationship where the deviation from the sustainable limit decrease until remaining very close to thelimit.

The steepness of the initial growth is largest for figure 4.3, and so is the deviation from the long-term sustainability limit of 6x10

4fish.


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5. A peculiar outcome is observed for higher values of the initial growth rate. Show this with aninitial growth rate of r=2.9. Explain the phenomenon.

One can start by writing two ordered pairs 4 4(1 10 , 2.9) , (6 10 , 1)


Trend line

y = -4E-05x + 3,28

11,11,21,31,4

1,51,61,71,81,9

22,12,22,32,42,52,62,72,82,9

0 10000 20000 30000 40000 50000 60000 70000

Fish population Un

Growthfactorr


The equation of the trend line is of the formy=mx + b:5

4 10 3.28y x

= + . Thus m=5

4 10

.This can be verified algebraically:

4 4

1.0 2.9

6 10 1 10

ym

= = =

53.8 10

Once again the use of technology is somewhat limiting because the value ofm is given to one significantfigure. Algebraically this can be amended.


5

5 4

5 4

( 3.8 10 )( )

2.9 ( 3.8 10 )(1 10 )

2.9 ( 3.8 10 )(1 10 ) 3.28

n n

n

r mu c

u c

c

c

= +

= +

= +

= =


= + {10}



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1

5

5

1

5

5 2

3.8 10 3.28

( 3.8 10 3.28)

( 3.8 10 )( )( ) 3.28

( 3.8 10 )( ) 3.28

n n

n n

n n n

n n n

n n

u ru

r u

u u u

u u u

u u

+

+

=

= +

= +

= +

= +


1 ( 3.8 10 )( ) 3.28n n nu u u

+= + {11}

Table 4.1.Growth factorr=2.9. The population of fish in a lake over a time range of 20 years estimated using the logistic function model{11}. The interval of calculation is 1 year.


1 1.0 410 11 7.07 410 2 2.9 410 12 4.19 410 3

6.32

4

10

13

7.07

4

10

4 5.56 410 14 4.19 410 5 6.49 410 15 7.07 410 6 5.28 410 16 4.19 410 7 6.73 410 17 7.07 410 8 4.87 410 18 4.19 410 9 6.96 410 19 7.07 410 10 4.42 410 20 4.19 410



-4300

700

5700

10700

15700

20700

25700

30700

35700

40700

45700

50700

55700

60700

65700

70700

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

Year

PopulationU

n+1/fish


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With a considerably large growth rate, r=2.9, the long-term sustainable limit is reached afteralready 3 years, and since this limit cannot be surpassed over a long time, the fourth year is

characterized by a drop in population (ie. the death of roughly thirty thousand fish), followed by a

demographic explosion. This continues on until two limits are reached, with a maximum and aminimum value. Hence a dynamic system is established. Instead of an asymptotic behaviour it goesfrom the maximum to the minimum [between the very low (4.19x10

4) and the excessively high

(7.07x104)].

As previously explained, the population cannotremain that high and must thus stabilize itself, yet

this search for stability only leads to another extreme value. The behaviour of the population isopposite to that of an equilibrium in the population (rate of birth=rate of death) as the rates are first

very high, but are then very low in the next year. This means that in year 11 the rate of deaths is veryhigh (or that of birth very low, or possibly both) and hence the population in year 12 is significantly

lower.Another aspect to consider when explaining this movement of the extrema is the growth factor.

Once the two limits are established rranges as follows: [0.593 , 1.69]r

and it attains the valuer=0.593 when the population is at its maximum going towards its minimum, and the value r=0.1.69when the population is at its minimum going towards its maximum.

6. Once the fish population stabilizes, the biologist and the regional managers see the commercialpossibility of an annual controlled harvest. The difficulty would be to manage a sustainableharvest without depleting the stock. Using the first model encountered in this task with r=1.5,

determine whether it would be feasible to initiate an annual harvest of 35 10 fish after a stable

population is reached. What would be the new, stable fish population with an annual harvest of

this size?

From figure 3.1 it could be said that the population stabilizes at around year 14 (due to a round-off). Most correctly this occurs in year 19 where the population strictly is equal to the long-term

sustainable limit. Nevertheless, if one began a harvest of 35 10 fish at any of these years(depending on what definition of stable is employed), one could say that un+1 is reduced by

35 10 , so that when one is calculating the growth of the population at the end of year 1, one takes

into account that 35 10 fish has been harvested. Mathematically this is shown as:5 2 3

1 ( 1 10 )( ) 1.6( ) 5 10n n nu u u

+ = + {12}

Table 6.1. The population of fish in a lake over a time range of 20 years estimated using the logistic function model {12}. A harvest of3

5 10 fish is made per annum.Year (after attaining stability) Population Year Population

1 6.0 410 11 5.00 410 2 5.55 410 12 5.00 410 3 5.28 410 13 5.00 410 4 5.16 410 14 5.00 410 5 5.09 410 15 5.00 410


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6 5.05 410 16 5.00 410 7 5.03 410 17 5.00 410 8 5.01 410 18 5.00 410 9 5.01 410 19 5.00 410 10 5.01 410 20 5.00 410

Estimated magnitude of population of fish of a hydrolectric project

during the first 20 years by means of the logistic function model

Un+1 {12} considering a harvest of 5000 fish per annum.

48000

50000

52000

54000

56000

58000

60000

62000

0 2 4 6 8 10 12 14 16 18 20Year

PopulationUn+1

/fish

Figure 6.1. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years using logistic function

model {12}. The model considers an annual harvest of 35 10 fish. The stable fish population is 5.0410 .

The behaviour observed in figure 6.1 resembles an inverse square relationship, and although there isno physical asymptote, the curve appears to approach some y-value, until it actually attains the y-

coordinate 5.0 410 . It is feasible to initiate an annual harvest of 35 10 fish afterattaining a stablepopulation (when that occurs has already been discussed). From the graph one can conclude that the

annual stable fish population with an annual harvest of 35.0 10 fish is 5.0x104.

7. Investigate other harvest sizes. Some annual harvests will cause the populations to die out.Illustrate your findings graphically.

a. Consider a harvest of 7.5 310 . Then the logistic function model is:5 2 3

1 ( 1 10 )( ) 1.6( ) 7.5 10n n nu u u

+ = + {13}

Table 7.1. The population of fish in a lake over a time range of 20 years estimated using the logistic function model {13}. A harvest of

7.5 310 fish is made per annum.

Year (afterattaining stability)

Population Year Population


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1 6.0 410 16 4.24 410 2 5.25 410 17 4.23 410 3 4.89 410 18 4.23 410 4 4.69 410 19 4.23 410 5 4.55 410 20 4.23 410

6 4.46 410 21 4.23 410 7 4.40 410 22 4.23 410 8 4.35 410 23 4.23 410 9 4.32 410 24 4.23 410 10 4.30 410 25 4.23 410 11 4.28 410 26 4.23 410 12 4.26 410 27 4.23 410 13 4.25 410 28 4.23 410 14 4.25 410 29 4.23 410 15

5.24

4

10

30

4.22

4

10



considering a harvest of 7500 fish per annum.

40000

42000

44000

46000

48000

50000

52000

54000

56000

58000

60000

62000

0 2 4 6 8 10 12 14 16 18 20Year

PopulationUn+1/fish

Figure 7.1. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years using logistic

function model {13}. The model considers an annual harvest of37.5 10 fish. The stable fish population is 4.97 410 .

From figure 7.1 one can observe this behaviour resembling a hyperbolic behaviour (x-1

). One can

conclude that it is feasible to initiate an annual harvest of 7.5 310 fish afterattaining a stable population


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(when that occurs has already been discussed). The curve approaches (and becomes) the value of4.22x104

fish, which is the annual stable fish population with an annual harvest of 7.5 310 fish.

b. Consider a harvest of 41 10 fish. Then the logistic function model is:5 2 4

1 ( 1 10 )( ) 1.6( ) 1 10n n nu u u

+ = + {14}

Table 7.2. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {14}. A harvest of1x104 fish is made per annum.

Year (after

attaining stability)


1 6.00 410 16 2.61 410 2 5.00 410 17 2.49 410 3 4.50 410 18 2.37 410 4 4.18 410 19 2.23 410 5 3.94 410 20 2.07 410 6 3.75 410 21 1.88 410 7 3.59 410 22 1.65 410 8 3.46 410 23 1.37 410 9 3.34 410 24 1.00 410 10 3.24 410 25 5.07 310 11 3.12 410 26 -2.15 310 12 3.02 410 27 -

13 2.92 410 28 -

14 2.82 410 29 -

15 2.72 410 30 -


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considering a harvest of 10,000 fish per annum.

0

10000

20000

30000

40000

50000

60000

70000

0 5 10 15 20 25

Year

PopulationUn+1/fish

Figure 7.2. Graphical plot of the fish population of the hydroelectric project on an interval of 25 years using the logistic

function model {14}. The model considers an annual harvest of 1x104 fish. A population collapse occurs after year 25.

From figure 7.2 one can observe the chronic depletion of the fish population as a consequence ofexcessive harvest. If one continues with the model into year 26 the population turns negative, and this

means the death of the last fish in the lake. Thus it is not feasible to harvest above 1 410 fish per annum,

and one could impose a preliminary condition when harvesting H fish p.a.:41 10H

8. Find the maximum annual sustainable harvest.When harvesting and a stable population are attained, one could argue that the rate of harvesting

equals the rate of growth of the population. But since one is not considering a differential equation,

one could instead say that the population un+1modelled in {12}, {13} and {14} can be generalizedwith model {15}:

5 2

1 ( 1 10 )( ) 1.6( )n n nu u u H

+ = + {15}

Because u1=6.0x104

(a stable value) then r=1. Therefore it is valid to write:

1(1)n nu u+ =

Thus if one consideredH=0 one could write {3} as following5 2

1 1 1( 1 10 )( ) 1.6( )n n nu u u

+ + += +

Now, if one began harvesting H fish, the equation would look like the following:


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5 2

1 1 1

5 2

1 1 1

( 1 10 )( ) 1.6( )

( 1 10 )( ) 1.6( ) 0

n n n

n n n

u u u H

u u u H

+ + +

+ + +

= +

+ =

5 21 1( 1 10 )( ) 0.6( ) 0n nu u H

+ + + = {16}

Model {16} is a quadratic equation which could be solved, if one knew the value of the constant

term H, yet in this case it serves as an unknown that must be found. This could be solved by recurringto an analysis of the discriminant of the quadratic equation:

5 2

1 1( 1 10 )( ) 0.6( ) 0n nu u H

+ + + =

2 5

1 5

0.6 0.6 4( 1 10 )( )

2( 1 10 )n

Hu

+

=

It is known that there can be

a. Two real roots (D>0, D )

b. One real root (D=0, D )

c. No real roots (D


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12 3.62 410 27 3.32 410 13 3.58 410 28 3.30 410 14 3.55 410 29 3.30 410 15 3.52 410 30 3.30 410

The population ultimately reaches the value 3.004

10 .

Estimated magnitude of population of fish of a hydrolectric project during the first 1x10^2 (order of

magnitude) years by means of the logistic function model Un+1 {17} considering a harvest of 90000 fish

per annum (Hmax).

0

10000

20000

30000

40000

50000

60000

70000

0 10 20 30 40 50 60 70 80 90 100

Year

PopulationUn+1

/fish

Un+1=3x10^3 fish after 1x10^2

(order of magnitude) years

Figure 8.1. Graphical plot of the fish population of the hydroelectric project on an interval of ca. 1x102 years (order of

magnitude) using the logistic function model {17}. The model considers an annual harvest of 9x103 fish. A stable population

(3x104) is reached after over 100 years (thus the order of magnitude).

9. Politicians in the area are anxious to show economic benefits from this project and wish to beginthe harvest before the fish population reaches its projected steady state. The biologist is called

upon to determine how soon fish may be harvested after the initial introduction of 10,000 fish.Again using the first model in this task, investigate different initial population sizes from which aharvest of 8,000 fish is sustainable.

The population sizes that was investigated, was:4

1 2 10u

This initial population is too small for it to be sustainable:


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8x103 fish is made per annum.

Year (after



1 1.95 410 11 1.65 410 2 1.94 410 12 1.57 410 3 1.93 410 13 1.47 410 4 1.91 410 14 1.33 410 5 1.89 410 15 1.16 410 6 1.87 410 16 9.16 310 7 1.84 410 17 5.82 310 8 1.81 410 18 9.68 210 9 1.77 410 19 -6.46 310 10 1.72 410 20 -1.88 410

Thus one can say, that the minimum initial population must be4

1 2 10u

a. Consider an initial population u1=2.5x104

fish.The equation could be formulated as follows:

5 2 3

1 1 1( 1 10 )( ) 1.6( ) 8 10n n nu u u

+ + + = + {19}




1 2.5 410 16 3.75 410 2 2.58 410 17 3.80 410 3 2.66 410 18 3.83 410 4 2.75 410 19 3.86 410 5 2.84 410 20 3.89 410 6 2.94 410 21 3.91 410 7 3.04 410 22 3.93 410 8 3.14 410 23 3.94 410 9 3.23 410 24 3.95 410 10

3.33

4

10

25

3.96

4

10

11 3.42 410 26 3.97 410 12 3.50 410 27 3.98 410 13 3.58 410 28 3.98 410 14 3.64 410 29 3.98 410 15 3.70 410 106 4.00 410


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Estimated magnitude of population of fish of a hydrolectric project during the first 106 years by means

of the logistic function model Un+1 {19} considering a harvest of 8000 fish per annum.

25000

26000

27000

28000

29000

30000

31000

32000

33000

34000

35000

36000

37000

38000

39000

40000

41000

0 20 40 60 80 100

Year

PopulationUn+1/fish


function model {19}. The model considers an annual harvest of 8x103 fish. A stable population (4x104) is reached.

b. Consider an initial population u1=3.0x104

fish.

The equation could be formulated as follows:5 2 3

1 1 1( 1 10 )( ) 1.6( ) 8 10n n nu u u

+ + + = + {19}




1 3.00 410 16 3.94 410 2 3.20 410 17 3.95 410 3 3.30 410 18 3.96 410 4 3.39 410 19 3.97 410 5 3.47 410 20 3.97 410 6 3.55 410 21 3.98 410

7 3.62 410 22 3.98 410 8 3.68 410 23 3.99 410 9 3.73 410 24 3.99 410 10 3.78 410 25 3.99 410 11 3.82 410 26 3.99 410 12 3.85 410 27 3.99 410 13 3.88 410 28 4.00 410 14 3.90 410 29 4.00 410


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22

15 3.92 410 30 4.00 410

Estimated magnitude of population of fish of a hydrolectric project during the first 30 years by means of

the logistic function model Un+1 {19} considering a harvest of 8000 fish per annum. Initial population is

30,000 fish

25000

26000

27000

28000

29000

30000

31000

32000

33000

34000

35000

36000

37000

38000

39000

40000

41000

0 5 10 15 20 25 30

Year

PopulationUn+1/fish

Figure 9.2. Graphical plot of the fish population of the hydroelectric project on an interval of 30 years using the logisticfunction model {19}. The model considers an annual harvest of 8x103 fish. A stable population (4x104) is reached.

c. Consider an initial population u1=3.5x104


5 2 3

1 1 1( 1 10 )( ) 1.6( ) 8 10n n nu u u

+ + + = + {19}


Year (after



1 3.50 410 16 3.98 410 2 3.58 410 17 3.98 410 3 3.64 410 18 3.98 410

4 3.704

10 19 3.994

10 5 3.75 410 20 3.99 410 6 3.80 410 21 3.99 410 7 3.83 410 22 3.99 410 8 3.86 410 23 3.99 410 9 3.89 410 24 3.99 410 10 3.91 410 25 4.00 410 11 3.93 410 26 4.00 410


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24/29


24

9 4.00 410 24 4.00 410 10 4.00 410 25 4.00 410 11 4.00 410 26 4.00 410 12 4.00 410 27 4.00 410 13 4.00 410 28 4.00 410

14 4.00 410 29 4.00 410 15 4.00 410 30 4.00 410



40,000 fish

39000

40000

41000

0 5 10 15 20 25 30

Year

PopulationUn+1/fish


function model {19}. The model considers an annual harvest of 8x103 fish. A stable population (4x104) is reached.

e. Consider an initial population u1=4.5x104 fish.

The equation could be formulated as follows:5 2 3

1 1 1( 1 10 )( ) 1.6( ) 8 10n n nu u u

+ + +

= + {19}




1 4.00 410 16 4.01 410 2 4.38 410 17 4.01 410 3 4.29 410 18 4.01 410 4 4.22 410 19 4.01 410


25/29


25

5 4.17 410 20 4.01 410 6 4.13 410 21 4.00 410 7 4.11 410 22 4.00 410 8 4.08 410 23 4.00 410 9 4.07 410 24 4.00 410

10 4.05 410 25 4.00 410 11 4.04 410 26 4.00 410 12 4.03 410 27 4.00 410 13 4.03 410 28 4.00 410 14 4.02 410 29 4.00 410 15 4.02 410 30 4.00 410



45,000 fish

39000

40000

41000

42000

43000

44000

45000

46000

0 5 10 15 20 25 30

Year

PopulationUn+1/fish


function model {19}. The model considers an annual harvest of 8x103

fish. A stable population (4x104) is reached.

f. Consider an initial population u1=6.0x104


5 2 3

1 1 1( 1 10 )( ) 1.6( ) 8 10n n nu u u

+ + + = + {19}


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26


8x103 fish is made per annum.Year (after



1 6.00 410 16 4.03 410 2 5.20 410 17 4.02 410 3

4.82

4

10

18

4.02

4

10

4 4.59 410 19 4.02 410 5 4.43 410 20 4.01 410 6 4.33 410 21 4.01 410 7 4.25 410 22 4.01 410 8 4.20 410 23 4.01 410 9 4.15 410 24 4.00 410 10 4.12 410 25 4.00 410 11 4.09 410 26 4.00 410 12 4.07 410 27 4.00 410

13 4.06 410 28 4.00 410 14 4.05 410 29 4.00 410 15 4.04 410 30 4.00 410


the logistic function model Un+1 {19} considering a harvest of 8000 fish per annum.

25000

27000

2900031000

33000

35000

37000

39000

41000

43000

45000

47000

49000

51000

53000

55000

57000

59000

61000

63000

0 5 10 15 20 25 30

Year

PopulationUn+1/fish

Figure 9.6. Graphical plot of the fish population of the hydroelectric project on an interval of 30 years using the logisticfunction model {19}. The model considers an annual harvest of 8x103 fish. A stable population (4x104) is reached.

One can conclude that the population reaches a positive, stable limit in the following interval:


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27

4 51 [2 10 ,1.4 10 [nu +

Also, as un+151.4 10 , the stable population instead of tending (and ultimately becoming) 44 10 , it

tends towards 42 10 , and when un+151.4 10 ,= the stable population is exactly 42 10 .

Thus it can be concluded, that when the initial population exceeds the stable limit, it will decrease in

magnitude, until reaching 44 10 fish, and if it lies below the stable population limit ( 44 10 ) it will

increase in magnitude until reaching stability. If one was to plot one initial population size of4

1 4 10u < versus4

1 4 10u > an equilibrium (resembling the chemical equilibrium of a reversible

reaction) is attained:


the logistic function model {19} for initial populations U1 =60000 and U1 *=30000 with an annual

harvest of 8000.

25000

27000

29000

31000

33000

35000

37000

39000

41000

43000

45000

47000

49000

51000

53000

55000

57000

59000

61000

63000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Year

PopulationUn+1/fish

Figure 9.7. Two initial population sizes attaining equilibrium at un+1= 44 10 for the model {19}.

What is the usefulness of such equilibrium? It tells us that if the politicians wanted to start such a

project with a set annual harvest, they could do this at the lowest cost by starting with a population of3x10

4fish instead of 6x10

4fish, obtain the same yearly harvest and keep the population alive. This is

the major socio-economic benefit of this study.Another point to this analysis is that, if one started with either of following values:

4

1

5

1

5

1

2 10

1.2 10

1.4 10

u

u

u

=

=

=

One would get to the same stable limit of 42 10 :5 4 2 4 3

1

4

( 1 10 )(2 10 ) 1.6(2 10 ) 8 10

2 10

nu

+ = +

=


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28

5 5 2 5 3

1

4

5 5 2 5 3

1

4

( 1 10 )(1.2 10 ) 1.6(1.2 10 ) 8 10

2 10

( 1 10 )(1.4 10 ) 1.6(1.4 10 ) 8 10

2 10

n

n

u

u

+

+

= +

=

= +

=

When graphing this one gets the follow figure:


the logistic function model {19} for initial populations U1 =20000 , U1 *=120000 and U1 **=140000

with an annual harvest of 8000.

15000

25000

35000

45000

55000

65000

75000

85000

95000

105000

115000

125000

135000

145000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Year

PopulationUn+1/fish

Figure 9.8. Three initial population sizes developing throughout 30 years time according to the model {19}.

Thus by starting the project with 2x104 one could (theoretically) maintain a stable population and get afairly high harvest out of it.

Conclusion

The logistic function {3} has been applied throughout the task and conclusions can be made about thelimitations of the model.

The model is strictly based on the assumption that r will remain constant at a value of r=1.5 until the

stable population limit is reached. Thus were the population to be influenced by some external factor, themodel would not take this into account and the growth of the population would undergo a discrepancy.

One could argue that the following assumptions were made when constructing the model:

The fish are ovoviviparous and carry the egg until it hatches1.

The fish population is somewhat homogenous.

1 Fish reproduction. < http://lookd.com/fish/reproduction.html> (online). Visited on 21. December 2006


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Their rate of reproduction is simplified and expressed as a growth factor instead of a rate of

change

Two cases could occur:a. An external factor increases the mortality of the population.

b. The fish reproduced at a higher rate than the model predicts.

Also, one has not taken into account that despite the model having the long-term sustainability limit atabout6x10

4, several factors could change this:

Human action such as contamination of the habitat.

Other aquatic vertebrates2

that might be positioned higher in the food chain, and could disturbthe stability by consuming the studied fishs own source of food.

Season changes

Global warmingThus it would be wise to introduce a term into the model that takes into the account the long-term

sustainability limit, which in a non-ideal situation probably would change.

The model predicts a few interesting things: Overpopulating the lake in the first year leads to the death of all fish by the end of year one. The

model is built foru1=1x104 fish, yet say one introduced one million fish in the lake (ignoring the

capacity of the lake):5 2

1 1 1 1

5 6 2 6 6

( 1 10 )( ) 1.6( )

( 1 10 )(1 10 ) 1.6(1 10 ) 8.4 10

u u u

fish

+

= +

= + =

Populations that reproduce at a very high rate (ie. r=2.9) result in a dynamic limit (with two valuesinstead of a stable limit with only one value) of fluctuations between the extrema, first from themaximum towards the minimum in only one year. This opposes the asymptotic behaviour

predicted when the growth factor is smaller.

When harvesting is introduced into a stable population (in this case at u1=6x104) only harvests below9x10

3fish may take place unless extinction of the population is desired.

When the harvesting is set at constant value of 8x103, initial populations below a specific value

will grow in magnitude until approaching that value, while initial populations above that value (yetnot above a certain limit as overpopulation causes death as well) lead to a decrease in magnitude

until approaching the same limit.

mathinvestigationii - logistic model

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