Hence the equation of the linear growth factor is:
r = −3.0 ×10−5 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 ×10−5 u
+ 2.8
n n
∴ u = (−3.0 ×10−5 u
+ 2.8)u
n+1 n n
= (−3.0 × 10−5 )(u
)(u
) + 2.8u
The logistic function model for un+1 is:
n n n
= (−3.0 × 10−5 )(u 2 ) + 2.8u
−5 2
un+1 = (−3.0 ×10 )(un
) + 2.8un
{9}
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
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 factor r=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 (yet due 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.
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 {5}
65000
60000
55000
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year
Figure 4.1. Growth factor r=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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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
Table 4.2. Growth factor r=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.
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 {7}
65000
60000
55000
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year
Figure 4.2. Growth factor r=2.3. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years using logistic function model {7}.
Table 4.3. Growth factor r=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.
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
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 {9}
65000
60000
55000
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year
Figure 4.3. Growth factor r=2.5. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years using logistic function model {9}.
The logistic function models attain greater values for m and c when the growth factor r increases in magnitude. As a consequence, the initial steepness of graphs 4.1, 4.2 and 4.3 increases, and thus the growth 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 the limit.
The steepness of the initial growth is largest for figure 4.3, and so is the deviation from the long- term sustainability limit of 6x104 fish.
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
5. A peculiar outcome is observed for higher values of the initial growth rate. Show this with an initial growth rate of r=2.9. Explain the phenomenon.
One can start by writing two ordered pairs (1×104 , 2.9) , (6 × 104 , 1)
2,9
2,8
2,7
2,6
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
Trend line
y = -4E-05x + 3,28
0 10000 20000 30000 40000 50000 60000 70000
Fish population Un
Figure 5.1. 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:
This can be verified algebraically:
y = −4 × 10−5 x + 3.28 . Thus m=
−4 × 10−5 .
m = Δy = Δx
1.0 − 2.9
6 × 104 − 1× 104
= −3.8 ×10−5
Once again the use of technology is somewhat limiting because the value of m is given to one significant figure. Algebraically this can be amended.
The linear growth factor is said to depend upon un and thus a linear equation can be written:
rn = mun + c
= (−3.8 ×10−5 )(u
) + c
2.9 = (−3.8 ×10−5 )(1×104 ) + c
⇒ c = 2.9 − (−3.8 × 10−5 )(1× 104 ) = 3.28
Hence the equation of the linear growth factor is:
r = −3.8 ×10−5 u
+ 3.28
{10}
n n
Using equations {1} and {2}, one can find the equation for un+1:
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
un+1 = run
r = −3.8 ×10−5 u
+ 3.28
n n
∴ un+1
= (−3.8 × 10−5 u
n + 3.28)un
= (−3.8 ×10−5 )(u
)(u
) + 3.28u
The logistic function model for un+1 is:
n n n
= (−3.8 ×10−5 )(u 2 ) + 3.28u
−5 2
un+1 = (−3.8 × 10 )(un
) + 3.28un
{11}
Table 4.1. Growth factor r=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.
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 {11}
70700
65700
60700
55700
50700
45700
40700
35700
30700
25700
20700
15700
10700
5700
700
-4300 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year
12
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
Figure 5.2. Growth factor r=2.9. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years using logistic function model {11}.
With a considerably large growth rate, r=2.9, the long-term sustainable limit is reached after already 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 a minimum value. Hence a dynamic system is established. Instead of an asymptotic behaviour it goes
from the maximum to the minimum [between the very low (4.19x104) and the excessively high
(7.07x104)].
As previously explained, the population cannot remain that high and must thus stabilize itself, yet
this search for stability only leads to another extreme value. The behaviour of the population is opposite 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 very high (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 r ranges as follows: r ∈ [0.593 , 1.69] and it attains the value r=0.593 when the population is at its maximum going towards its minimum, and the value r=0.1.69
when the population is at its minimum going towards its maximum.
6. Once the fish population stabilizes, the biologist and the regional managers see the commercial possibility of an annual controlled harvest. The difficulty would be to manage a sustainable harvest 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 5 ×103 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 5 ×103
fish at any of these years
(depending on what definition of ‘stable’ is employed), one could say that un+1 is reduced by
5 ×103 , so that when one is calculating the growth of the population at the end of year 1, one takes
into account that 5 ×103
fish has been harvested. Mathematically this is shown as:
−5 2 3
un+1 =
(−1× 10 )(un )
+ 1.6(un )
− 5 ×10
{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 of
5 ×103 fish is made per annum.
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
62000
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.
60000
58000
56000
54000
52000
50000
48000
0 2 4 6 8
Y1e0ar
12 14 16 18 20
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 5 × 103 fish. The stable fish population is 5.0×104 .
The behaviour observed in figure 6.1 resembles an inverse square relationship, and although there is no physical asymptote, the curve appears to approach some y-value, until it actually attains the y-
coordinate 5.0×104 . It is feasible to initiate an annual harvest of 5 ×103
fish after attaining a stable
population (when that occurs has already been discussed). From the graph one can conclude that the annual stable fish population with an annual harvest of 5.0 ×103 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 7.5×103 . Then the logistic function model is:
−5 2 3
un+1 =
(−1× 10 )(un )
+ 1.6(un )
− 7.5 ×10
{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 ×103 fish is made per annum.
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
62000
Estimated magnitude of population of fish of a hydrolectric project during the first 20 years by means of the logistic function model Un+1 {13} considering a harvest of 7500 fish per annum.
60000
58000
56000
54000
52000
50000
48000
46000
44000
42000
40000
0 2 4 6 8 10 12 14 16 18 20
Year
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 of 7.5 ×103 fish. The stable fish population is 4.97×104 .
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 ×103 fish after attaining a stable population
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
(when that occurs has already been discussed). The curve approaches (and becomes) the value of 4.22x104
fish, which is the annual stable fish population with an annual harvest of 7.5×103 fish.
b. Consider a harvest of 1× 104 fish. Then the logistic function model is:
−5 2 4
un+1 =
(−1× 10 )(un )
+ 1.6(un )
− 1× 10
{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 of
1x104 fish is made per annum.
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
70000
Estimated magnitude of population of fish of a hydrolectric project during the first 25 years by means of the logistic function model Un+1 {14} considering a harvest of 10,000 fish per annum.
60000
50000
40000
30000
20000
10000
0
0 5 10 15 20 25
Y ear
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 of excessive 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 ×104
and one could impose a preliminary condition when harvesting H fish p.a.:
H ≤ 1× 104
fish per annum,
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+1 modelled in {12}, {13} and {14} can be generalized with model {15}:
un+1 =
(−1× 10−5 )(u )2
+ 1.6(un ) − H
{15}
Because u1=6.0x104 (a stable value) then r=1. Therefore it is valid to write:
un+1 = (1)un
Thus if one considered H=0 one could write {3} as following
−5 2
un+1 = (−1× 10 )(un +1 )
+ 1.6(un+1 )
Now, if one began harvesting H fish, the equation would look like the following:
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
un+1 =
(−1× 10−5 )(u
n+1
)2 + 1.6(u
n+1 ) − H
⇒ ࣰ࣮(−1×10−5 )(u
n+1
)2 + 1.6(u
n+1
)ࣹࣻ − u
n+1
− H = 0
⇒ (−1×10−5 )(u )2
+ 0.6(un+1 )
− H = 0
{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 recurring to an analysis of the discriminant of the quadratic equation:
−5 2
(−1×10 )(un+1 )
+ 0.6(un+1 )
− H = 0
⇒ un+1 =
−0.6 ±
0.62 − 4(−1×10−5 )(−H )
2(−1× 10−5 )
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<0, D ∈ )
For a solution (exact value for the maximum value for H) one could consider a single real root:
−0.6 ±
0.62 − 4(−1×10−5 )(−H )
−0.6 ± D
un+1 =
2(−1× 10−5 )
=
2(−1×10−5 )
⇒ D = 0.62 − 4(−1×10−5 )(−H ) = 0
⇒ H ࣫
−0.62
ࣶ = 9000 = 9 ×103
= − ࣬ −4(−1× 10−5 ) ࣷ
Thus, the maximum value of harvest, that still preserves the fish population at a stable value is H=9000.
This value results in the following population development:
−5 2 3
un+1 =
(−1× 10 )(un+1 )
+ 1.6(un+1 )
− 9 × 10
{17}
Table 8.1. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {14}. A harvest of
1x104 fish is made per annum.
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
The population ultimately reaches the value 3.00 ×104 .
70000
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 (H max ).
60000
50000
40000
30000
20000
Un+1=3x10^3 fish after 1x10^2 (order of magnitude) years
10000
0
0 10 20 30 40 50 60 70 80 90 100
Year
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 begin the 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 a harvest of 8,000 fish is sustainable.
The population sizes that was investigated, was:
u1 ≤ 2 ×10
This initial population is too small for it to be sustainable:
19
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
Table 9.0. The population of fish in a lake over a time range of 20 years estimated using the logistic function model {14}. A harvest of
8x103 fish is made per annum.
Thus one can say, that the minimum initial population must be
u1 ≥ 2 ×10
a. Consider an initial population u1=2.5x104 fish. The equation could be formulated as follows:
−5 2 3
un+1 =
(−1× 10 )(un+1 )
+ 1.6(un+1 )
− 8 ×10
{19}
Table 9.1. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {14}. A harvest of
1x104 fish is made per annum.
20
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
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.
41000
40000
39000
38000
37000
36000
35000
34000
33000
32000
31000
30000
29000
28000
27000
26000
25000
0 20 40 60 80 100
Year
Figure 9.1. Graphical plot of the fish population of the hydroelectric project on an interval of 106 years using the logistic 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
un+1 =
(−1× 10 )(un+1 )
+ 1.6(un+1 )
− 8 ×10
{19}
Table 9.2. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {19}. A harvest of
8x103 fish is made per annum.
21
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
41000
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
40000
39000
38000
37000
36000
35000
34000
33000
32000
31000
30000
29000
28000
27000
26000
25000
0 5 10 15 20 25 30
Year
Figure 9.2. Graphical plot of the fish population of the hydroelectric project on an interval of 30 years using the logistic function 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 fish. The equation could be formulated as follows:
−5 2 3
un+1 =
(−1× 10 )(un+1 )
+ 1.6(un+1 )
− 8 ×10
{19}
Table 9.3. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {19}. A harvest of
8x103 fish is made per annum.
22
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
41000
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
35,000 fish
40000
39000
38000
37000
36000
35000
34000
0 5 10 15 20 25 30
Year
Figure 9.3. Graphical plot of the fish population of the hydroelectric project on an interval of 106 years using the logistic function model {19}. The model considers an annual harvest of 8x103 fish. A stable population (4x104) is reached.
d. Consider an initial population u1=4.0x104 fish. The equation could be formulated as follows:
−5 2 3
un+1 =
(−1× 10 )(un+1 )
+ 1.6(un+1 )
− 8 ×10
{19}
Table 9.4. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {19}. A harvest of
8x103 fish is made per annum.
23
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
41000
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
40,000 fish
40000
39000
0 5 10 15 20 25 30
Year
Figure 9.4. Graphical plot of the fish population of the hydroelectric project on an interval of 30 years using the logistic 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
un+1 =
(−1× 10 )(un+1 )
+ 1.6(un+1 )
− 8 ×10
{19}
Table 9.5. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {19}. A harvest of
8x103 fish is made per annum.
24
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
46000
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
45,000 fish
45000
44000
43000
42000
41000
40000
39000
0 5 10 15 20 25 30
Year
Figure 9.5. Graphical plot of the fish population of the hydroelectric project on an interval of 30 years using the logistic 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 fish. The equation could be formulated as follows:
−5 2 3
un+1 =
(−1× 10 )(un+1 )
+ 1.6(un+1 )
− 8 ×10
{19}
25
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
Table 9.6. The population of fish in a lake over a time range of 30 years estimated using the logistic function model {19}. A harvest of
8x103 fish is made per annum.
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.
63000
61000
59000
57000
55000
53000
51000
49000
47000
45000
43000
41000
39000
37000
35000
33000
31000
29000
27000
25000
0 5 10 15 20 25 30
Year
Figure 9.6. Graphical plot of the fish population of the hydroelectric project on an interval of 30 years using the logistic function 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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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
Also, as un+1 → 1.4 ×105 ,
u ∈ [2 × 104 ,1.4 × 105 [
the stable population instead of tending (and ultimately becoming)
4 × 104 , it
tends towards 2 × 104 , and when un+1 = 1.4 × 105 , the stable population is exactly 2 ×104 .
Thus it can be concluded, that when the initial population exceeds the stable limit, it will decrease in
magnitude, until reaching
4 ×104 fish, and if it lies below the stable population limit ( 4 ×104 ) it will
increase in magnitude until reaching stability. If one was to plot one initial population size of
u1 • 4 ×10
versus u1 • 4 ×10
an equilibrium (resembling the chemical equilibrium of a reversible
reaction) is attained:
63000
61000
59000
57000
55000
53000
51000
49000
47000
45000
43000
41000
39000
37000
35000
33000
31000
29000
27000
25000
Estimated magnitude of population of fish of a hydrolectric project during the first 30 years by means of the logistic function model {19} for initial populations U 1 =60000 and U 1 *=30000 with an annual harvest of 8000.
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
Figure 9.7. Two initial population sizes attaining equilibrium at un+1= 4 × 104 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 of
3x104 fish instead of 6x104 fish, 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:
One would get to the same stable limit of 2 ×104 :
u = 2 × 104
u = 1.2 × 105
u = 1.4 × 105
un+1 =
(−1× 10−5 )(2 × 104 )2 + 1.6(2 × 104 )
− 8 ×103
= 2 ×104
27
IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
un+1 =
(−1× 10−5 )(1.2 × 105 )2 + 1.6(1.2 × 105 )
− 8 × 103
= 2 ×104
un+1 =
(−1× 10−5 )(1.4 × 105 )2 + 1.6(1.4 × 105 )
− 8 × 103
= 2 ×104
When graphing this one gets the follow figure:
Estimated magnitude of population of fish of a hydrolectric project during the first 30 years by means of the logistic function model {19} for initial populations U 1 =20000 , U 1 *=120000 and U 1 **=140000
with an annual harvest of 8000.
145000
135000
125000
115000
105000
95000
85000
75000
65000
55000
45000
35000
25000
15000
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
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 a fairly high harvest out of it.
Conclusion
The logistic function {3} has been applied throughout the task and conclusions can be made about the limitations 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, the model 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. < > (online). Visited on 21. December 2006
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IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg – Christian Jorgensen
∙ 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 at
about 6x104, 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 disturb the stability by consuming the studied fish’s own source of food.
∙ Season changes
∙ Global warming
Thus 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 for u1=1x104 fish, yet say one introduced one million fish in the lake (ignoring the capacity of the lake):
u = (−1× 10−5 )(u )2 + 1.6(u )
1+1 1 1
= (−1× 10−5 )(1× 106 )2 + 1.6(1×106 ) = −8.4 × 106 fish
∙ Populations that reproduce at a very high rate (ie. r=2.9) result in a dynamic limit (with two values instead of a stable limit with only one value) of fluctuations between the extrema, first from the maximum 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 below
9x103 fish 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 (yet not above a certain limit as overpopulation causes death as well) lead to a decrease in magnitude
until approaching the same limit.
2 Biological term for fish: Wikipedia < Visited on 21. December 2006
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