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Math Portfolio Type II

Extracts from this document...

Introduction

Mathematics Portfolio

Type II

Creating a Logistic Model

Description

A geometric population model takes the formimage11.png 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: -

  1. 10,000 fishes are introduced into a lake
  2. The population increase if 10,000 fishes are introduced into the lake would be by 50% in the first year.
  3. 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 image12.png . Now, if we have to find out the ordered pair (u0,r0): -

  • It is mentioned in the description that unis 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 image27.pngimage27.png

            = 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)

...read more.

Middle

60323

6

57375

16

59737

7

61892

17

60207

8

58378

18

59382

9

61218

19

60133

20

59893

The above tabulated data can be represented by the following logistic function graph: -

image39.pngimage40.png

The table and graph confirm how higher growth rates above 2 lead to less stability in population. Here, the variance from the long term sustainable limit (which is 60000) is much more (64703 in the 3rd year) compared to when the initial growth rate was 2.3 or 2. The successive year sees a similar larger fall to 55573 in population compared to the other two cases. Consequently, the table confirms how a high initial growth rate leads to longer time for the population to stabilize at the long term sustainable limit. In this case, it is not attained in the first 20 years. Rather, upon further calculations we can find out that the population becomes stable at the long term sustainable limit in the 41st year compared to the 17th year when the initial growth rate is 2.3 .

Part 5

Let us see what happens when the initial growth rate is 2.9

The ordered pair (u0, r0) for the first pair when the predicted growth rate ‘r’ for the year is 2 will be (10000, 2.9).

(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.

image41.png

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

image42.pngimage43.png

y = (-3.8 x 10-5)x + 3.28

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 = (-3.8 x 10-5)un + 3.28

Now, to find the recursive logistic function, we shall substitute the value of rn calculated above in the following form un+1 = rn.un

un+1 = [(-3.8 x 10-5)un + 3.28]un

        = (-3.8 x 10-5)un2 + 3.28un

To find out the changes in the pattern, let us see the following table which shows the first 20 values of the population obtained according to the logistic function un+1= (-3 x 10-5)un2 + 2.8un just calculated: -

n+1

un+1image02.png

n+1

un+1

0

10000

10

70738

1

29000

11

41872

2

63162

12

70716

3

55572

13

41919

4

64922

14

70720

5

52779

15

41910

6

67261

16

70719

7

48701

17

41911

8

69611

18

70719

9

44187

19

41911

20

70719

The above tabulated data can be represented by the following logistic function graph: -

image44.pngimage45.png

image04.png

We can infer from the data in the table and the graph how a high initial growth rate of 2.9 can lead to fluctuating populations every year. The long term sustainable limit of 60,000 is crossed at the end of the 2nd year itself, becoming 63162. But, in such a case, the environment (in this case the lake) cannot withhold such a population and thus there are a lot of fishes either dying out or migrating in the subsequent year which reduces the population to 55572. Yet, again in the next year, due to high growth rate, the population reaches well above even 63162 to 64922. Such a fluctuating trend is observed in the succeeding years until the end of the 16th and 17th year during and after which the population fluctuates between 70719 and 41911.

Part 6

To initiate an annual harvest of 5000 fishes, let us first find out at what point the fish population stabilizes when the initial growth rate, r = 1.5.

The following table shows the growth in the population over the first 20 years taking r = 1.5 and thus making use of the following recursive relation as found out in Part 2:-

un+1 = (-1 x 10-5) un2 + 1.6un

In this relation, the starting population is the initial population u0 = 10000.

n+1

un+1image05.png

n+1

un+1

0

10000

10

59772

1

15000

11

59908

2

21750

12

59963

3

30069

13

59985

4

39069

14

59994

5

47246

15

59997

6

53272

16

59999

7

56856

17

59999

8

58643

18

59999

9

59439

19

59999

20

59999

Here, we see that the population reaches a stable point, when r =1 at the end of the 16th year. According to the question, we shall initiate the harvest of 5000 fishes every year from this point onwards. The following changes are made in the original recursive relation:-

  1. The starting population is taken to be 59999
  2. In the relation un+1 = (-1 x 10-5) un2 + 1.6un  , we subtract 5000 as this is the number which is removed every year. Thus, the new relation becomes

un+1 = (-1 x 10-5) un2 + 1.6un – 5000

It should be noted that although the start population or u0 is taken to be 59999, the starting year is the 16th year.

Entering these values in the RECUR mode of the GDC Casio CFX 9850GC Plus, we get the following table:-

n+1

un+1image06.png

n+1

un+1

16

59999

26

50041

17

54999

27

50024

18

52749

28

50014

19

51574

29

50008

20

50919

30

50005

21

50543

31

50003

22

50323

32

50001

23

50192

33

50001

24

50115

34

50000

25

50069

35

50000

image46.png

We see that there is a declining trend in the fish population with an annual harvest of 5000 fishes when the growth rate is 1.5. However, it should be noted that with this annual harvest, the population finally stabilizes in the 34th year when it becomes 50000.

The graph is as follows: -

image47.pngimage40.png

image07.png

Thus, we can conclude that it is feasible to initiate an annual harvest of 5000 fishes once the stable population of 59999 is reached in the 16th year with the growth rate being 1.5.

Initially, a declining trend in the fish population is observed as 5000 fishes are harvested every year. Yet, from the 34th year onwards, we observe that the new stable fish population becomes 50,000 with an annual feasible harvest of 5000 fishes.

Part 7

Taking the same model in which the growth rate r = 1.5, we shall investigate other harvest sizes.

n+1

un+1image02.png

n+1

un+1

0

10000

10

59772

1

15000

11

59908

2

21750

12

59963

3

30069

13

59985

4

39069

14

59994

5

47246

15

59997

6

53272

16

59999

7

56856

17

59999

8

58643

18

59999

9

59439

19

59999

20

59999

  • When the harvest size = 2500 fishes

We shall initiate the harvest of 2500 fishes after the 16th year when the population stabilizes at 59999. The following changes are made in the original recursive relation:-

  1. The starting population is taken to be 59999
  2. In the relation un+1 = (-1 x 10-5) un2 + 1.6un  , we subtract 2500 as this is the number which is removed every year. Thus, the new relation becomes

un+1 = (-1 x 10-5) un2 + 1.6un – 2500

Entering these values in the RECUR mode of the GDC Casio CFX 9850GC Plus, we get the following table:-

n+1

un+1image06.png

n+1

un+1

16

59999

26

55498

17

57499

27

55496

18

56437

28

55495

19

55948

29

55495

20

55715

30

55495

21

55602

31

55495

22

55547

32

55495

23

55520

33

55495

24

55507

34

55495

25

55501

35

55495

image46.png

The graph is as follows: -

image49.pngimage40.png

We see that when the annual harvest is 2500, there is a declining trend which stabilizes quite early, that is from the 28th year onwards compared to when the harvest is 5000 fishes. Also, the new stable population, 55495 is higher compared to when the harvest if 5000 fishes.

Thus, we can conclude that it is also feasible to initiate an annual harvest of 2500 fishes and the new stable population would be achieved from the 28th year onwards as 55495.

  • When the harvest size = 7500

We shall initiate the harvest of 7500 fishes after the 16th year when the population stabilizes at 59999. The following changes are made in the original recursive relation:-

  1. The starting population is taken to be 59999
  2. In the relation un+1 = (-1 x 10-5) un2 + 1.6un  , we subtract 7500 as this is the number which is removed every year. Thus, the new relation becomes

un+1 = (-1 x 10-5) un2 + 1.6un – 7500

The table settings are as follows: -

image50.png

Entering these values in the RECUR mode of the GDC Casio CFX 9850GC Plus, we get the following table:-

n+1

un+1image06.png

n+1

un+1

n+1image06.png

un+1image06.png

n+1

un+1

16

59999

26

42774

36

42278

46

42249

17

52499

27

42642

37

42270

47

42248

18

48937

28

42544

38

42265

48

42248

19

46851

29

42470

39

42260

49

42248

20

45511

30

42415

40

42257

50

42248

21

44605

31

42374

41

42255

51

42248

22

43972

32

42342

42

42253

52

42247

23

43520

33

42319

43

42251

53

42247

24

43192

34

42301

44

42250

54

42247

25

42951

35

42288

45

42249

55

42247

...read more.

Conclusion

-5) un2 + 1.6un – 8000. Also, the starting population, u0 = 16000. This shall be entered in the GDC Casio CFX-9850GC Plus.

image101.pngimage14.png

In this case also, we find that the population dies put during the 7th year which means that  such a combination also is not sustainable.

  • Let us try with an initial population size of 19000 fishes and initiate a harvest of 8000 fishes from the 1st year itself.

The new recursive function becomes un+1 = (-1 x 10-5) un2 + 1.6un – 8000. Also, the starting population, u0 = 19000. This shall be entered in the GDC Casio CFX-9850GC Plus.

image15.pngimage16.png

The graph and GDC show that the population dies out during the 14th year which means that this combination is not sustainable.

  • When the initial population size is 20000 fishes and a harvest of 8000 fishes is initiated from the 1st year itself.

The new recursive function becomes un+1 = (-1 x 10-5) un2 + 1.6un – 8000. Also, the starting population, u0 = 20000. This shall be entered in the GDC Casio CFX-9850GC Plus.

image17.pngimage16.png

Using this combination, we have found out using the GDC that the population remains constant and stable starting from the 1st year itself. It neither depletes nor increases. Thus, this introductory fish population size of 20000 is the most sustainable for harvesting 8000 fishes.

  • To prove that an initial population of 20000 fishes is the most suitable and sustainable for a harvest of 8000 fishes every year, we shall investigate using u0 = 19999.

The new recursive function becomes un+1 = (-1 x 10-5) un2 + 1.6un – 8000. Also, the starting population, u0 = 19999. This shall be entered in the GDC Casio CFX-9850GC Plus.

image18.pngimage19.png

The GDC shows that in such a case, the population has a decreasing trend which slowly increases with increasing number of years and finally, dies out during the 52nd year. Hence, an initial population size of 19999 is not sustainable.

These results help us to infer that a minimum initial population size of 20000 fishes is required to sustain a harvest of 8000 fishes every year from the 1st year.

Candidate Name: - Sanchit Ladha

Candidate Session No.: -1070-006                

...read more.

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