Creating a logistic model Christian Jorgensen

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1 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Creatig a logistic model Christia Jorgese IB Diploma Programme IB Mathematics HL Portfolio type 2 Cadidate umber vvvvvvvvvv Iteratioal School of Helsigborg, Swede

2 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Theory A logistic model is expressed as: u ru = + {} The growth factor r varies accordig to u. If r= the the populatio is stable. Solutio. A hydroelectric project is expected to create a large lake ito which some fish are to be placed. A biologist estimates that if x fish were itroduced ito the lake, the populatio of fish would icrease by 50% i the first year, but the log-term sustaiability limit would be about 6 x. From the iformatio above, write two ordered pairs i the form (u 0, r 0 ), (u 0, r 0 ) where U =6x. Hece, determie the slope ad equatio of the liear growth i terms of U As U =6, the populatio i the lake is stable. Thus from the defiitio of a logistic model, r must equal to as u approaches the limit. If the growth i populatio of fish (iitially ) is 50% durig the first year, r must be equal to.5. Hece the ordered pairs are: (,.5), (6, ) Oe ca graph the two ordered pairs. It has bee requested to fid the growth factor i terms of U, ad thus oe should graph the populatio (x-axis) versus the growth factor (y-axis): Growth factor r 2,5 2, 2, 2,2 2, 2,9,8,7,6,5,,,2, Plot of populatio U versus the growth factor r Tredlie y = -E-05x +, Fish populatio U Figure.. Graphical plot of the fish populatio U versus the growth factor r of the logistic model u + = ru 2

3 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese The slope of the tred lie i figure is determied graphically. The equatio of the tred lie is of the form y=mx + b: y = x+.6. Thus m=. This ca be verified algebraically: y.0.5 m = = = x 6 The liear growth factor is said to deped upo u ad thus a liear equatio ca be writte: r = mu + c = ( )( u ) + c.5 = ( )( ) + c c =.5 ( )( ) =.6 Hece the equatio of the liear growth factor is: r = u +.6 {2} 2. Fid the logistic fuctio model for u + Usig equatios {} ad {2}, oe ca fid the equatio for u +: u = ru r + = u +.6 u = u + u + (.6) = u u + u ( )( )( ).6 2 = ( )( u ) +.6u The logistic fuctio model for u + is: 5 2 u = ( + )( u ) +.6u {}. Usig the model, determie the fish populatio over the ext 20 years ad show these values usig a lie graph Oe ca determie the populatio of the first 20 years just by kowig that u = ad u 2 =.5. For istace u is 5 2 u = ( )( u2 ) +.6u2 2 = ( )(.5 ) +.6(.5 ) = 2.8 Table.. The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {}. The iterval of calculatio is year. Populatio Populatio

4 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Populatio U / fish Estimated magitude of populatio of fish of a hydrolectric project durig the first 20 years by meas of the logistic fuctio model U Figure.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 20 years usig logistic fuctio model {}. The graph is asymptotic approachig 6.0x fish despite the years 9 ad 20 showig data rouded off to 6.0x.. The biologist speculates that the iitial growth rate may vary cosiderably. Followig the process above, fid ew logistic fuctio models for u + usig the iitial growth rates r=2, 2. ad 2.5. Describe ay ew developmets. a. For r=2. Oe ca write two ordered pairs (, 2.0), (6, ). The graph of the two ordered pairs is:

5 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Plot of populatio U versus the growth factor r Growth factor r 2,5 2, 2, 2,2 2, 2,9,8,7,6,5,,,2, y = -2E-05x + 2, Fish populatio U Figure.. Graphical plot of the fish populatio U versus the growth factor r of the logistic model u The equatio of the tred lie is of the form y=mx + b: y = 2 x Thus m= ca be verified algebraically: y m = = = 2 x 6 + = ru 2. This The liear growth factor is said to deped upo u ad thus a liear equatio ca be writte: r = mu + c = ( 2 )( u ) + c 2.0 = ( 2 )( ) + c c = 2.0 ( 2 )( ) = 2.2 Hece the equatio of the liear growth factor is: r = 2 u {} Usig equatios {} ad {2}, oe ca fid the equatio for u +: u = ru r + = u u = u + u + ( 2 2.2) = u u + u ( 2 )( )( ) = ( 2 )( u ) + 2.2u The logistic fuctio model for u + is: 5 2 u = ( 2 + )( u ) + 2.2u {5} b. For r=2.. Oe ca write two ordered pairs (, 2.), (6, ). The graph of the two ordered pairs is: 5

6 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Plot of populatio U versus the growth factor r Growth factor r 2,5 2, 2, 2,2 2, 2,9,8,7,6,5,,,2, y = -E-05x + 2, Fish populatio U Figure.2. Graphical plot of the fish populatio U versus the growth factor r of the logistic model u The equatio of the tred lie is of the form y=mx + b: y = x Thus m= 2. This ca be verified algebraically: y.0 2. m = = = 2.6 x 6 The differece surges because the program used to graph rouds m up to oe sigificat figure. The liear growth factor is said to deped upo u ad thus a liear equatio ca be writte: r = mu + c = ( 2.6 )( u ) + c 2. = ( 2.6 )( ) + c c = 2. ( 2.6 )( ) = 2.56 Hece the equatio of the liear growth factor is: r = 2.6 u {6} + = ru Usig equatios {} ad {2}, oe ca fid the equatio for u +: u = ru r + = u u = u + u + ( ) = u u + u ( 2.6 )( )( ) = ( 2.6 )( u ) u The logistic fuctio model for u + is: 5 2 u = ( 2.6 )( u ) u {7} c. For r=2.5. Oe ca write two ordered pairs (, 2.5), (6, ). The graph of the two ordered pairs is: 6

7 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Plot of populatio U versus the growth factor r Growth factor r 2,5 2, 2, 2,2 2, 2,9,8,7,6,5,,,2, y = -E-05x + 2, Fish populatio U Figure.. Graphical plot of the fish populatio U versus the growth factor r of the logistic model u + = ru The equatio of the tred lie is of the form y=mx + b: y = x Thus m= ca be verified algebraically: y m = = =.0 x 6. This The liear growth factor is said to deped upo u ad thus a liear equatio ca be writte: r = mu + c = (.0 )( u ) + c 2.5 = (.0 )( ) + c c = 2.5 (.0 )( ) = 2.8 Hece the equatio of the liear growth factor is: r =.0 u {8} Usig equatios {} ad {2}, oe ca fid the equatio for u +: u = ru r + = u u = u + u + (.0 2.8) = u u + u (.0 )( )( ) = (.0 )( u ) + 2.8u The logistic fuctio model for u + is: 5 2 u = (.0 + )( u ) + 2.8u {9} 7

8 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese The fidigs for a, b ad c eables oe to calculate the magitude of the populatio for three studies with differet growth factor. These results are show schematically ad graphically: Table.. Growth factor r=2.0 The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {5}. The iterval of calculatio is year. Please ote that from year 6 ad owards the populatio resoates above ad below the limit (yet due to the roudig up of umbers this is ot observed), ad it fially stabilizes i year 7 (ad owards) where the populatio is exactly equal to the sustaiable limit. Populatio Populatio Populatio U + / fish Estimated magitude of populatio of fish of a hydrolectric project durig the first 20 years by meas of the logistic fuctio model U + {5} Figure.. Growth factor r=2.0. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 20 years usig logistic fuctio model {5}. 8

9 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Table.2. Growth factor r=2. The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {7}. The iterval of calculatio is year. Populatio Populatio Populatio U + / fish Estimated magitude of populatio of fish of a hydrolectric project durig the first 20 years by meas of the logistic fuctio model U + {7} Figure.2. Growth factor r=2.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 20 years usig logistic fuctio model {7}. Table.. Growth factor r=2.5. The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {9}. The iterval of calculatio is year. Populatio Populatio

10 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Populatio U + / fish Estimated magitude of populatio of fish of a hydrolectric project durig the first 20 years by meas of the logistic fuctio model U + {9} Figure.. Growth factor r=2.5. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 20 years usig logistic fuctio model {9}. The logistic fuctio models attai greater values for m ad c whe the growth factor r icreases i magitude. As a cosequece, the iitial steepess of graphs.,.2 ad. icreases, ad thus the growth of the populatio icrease. Also, as these values icrease the populatio (accordig to this model) goes above the sustaiable limit but the drops below the subsequet year so as to stabilize. There is o loger a asymptotic relatioship (goig from {5} to {9}) but istead a stabilizig relatioship where the deviatio from the sustaiable limit decrease util remaiig very close to the limit. The steepess of the iitial growth is largest for figure., ad so is the deviatio from the logterm sustaiability limit of 6x fish.

11 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese 5. A peculiar outcome is observed for higher values of the iitial growth rate. Show this with a iitial growth rate of r=2.9. Explai the pheomeo. Oe ca start by writig two ordered pairs (, 2.9), (6, ) Growth factor r 2,9 2,8 2,7 2,6 2,5 2, 2, 2,2 2, 2,9,8,7,6,5,,,2, Plot of populatio U versus the growth factor r Tred lie y = -E-05x +, Fish populatio U Figure 5.. Graphical plot of the fish populatio U versus the growth factor r of the logistic model u + = ru The equatio of the tred lie is of the form y=mx + b: y = x+.28. Thus m=. This ca be verified algebraically: y m = = =.8 x 6 Oce agai the use of techology is somewhat limitig because the value of m is give to oe sigificat figure. Algebraically this ca be ameded. The liear growth factor is said to deped upo u ad thus a liear equatio ca be writte: r = mu + c = (.8 )( u ) + c 2.9 = (.8 )( ) + c c = 2.9 (.8 )( ) =.28 Hece the equatio of the liear growth factor is: r =.8 u +.28 {} Usig equatios {} ad {2}, oe ca fid the equatio for u +:

12 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese u r + = ru = u u = u + u + (.8.28) = u u + u (.8 )( )( ).28 2 = (.8 )( u ) +.28u The logistic fuctio model for u + is: 5 2 u = (.8 + )( u ) +.28u {} Table.. Growth factor r=2.9. The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {}. The iterval of calculatio is year. Populatio Populatio Estimated magitude of populatio of fish of a hydrolectric project durig the first 20 years by meas of the logistic fuctio model U + {} Populatio U + / fish

13 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Figure 5.2. Growth factor r=2.9. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 20 years usig logistic fuctio model {}. With a cosiderably large growth rate, r=2.9, the log-term sustaiable limit is reached after already years, ad sice this limit caot be surpassed over a log time, the fourth year is characterized by a drop i populatio (ie. the death of roughly thirty thousad fish), followed by a demographic explosio. This cotiues o util two limits are reached, with a maximum ad a miimum value. Hece a dyamic system is established. Istead of a asymptotic behaviour it goes from the maximum to the miimum [betwee the very low (.9x ) ad the excessively high (7.07x )]. As previously explaied, the populatio caot remai that high ad must thus stabilize itself, yet this search for stability oly leads to aother extreme value. The behaviour of the populatio is opposite to that of a equilibrium i the populatio (rate of birth=rate of death) as the rates are first very high, but are the very low i the ext year. This meas that i year the rate of deaths is very high (or that of birth very low, or possibly both) ad hece the populatio i year 2 is sigificatly lower. Aother aspect to cosider whe explaiig this movemet of the extrema is the growth factor. Oce the two limits are established r rages as follows: r [0.59,.69] ad it attais the value r=0.59 whe the populatio is at its maximum goig towards its miimum, ad the value r=0..69 whe the populatio is at its miimum goig towards its maximum. 6. Oce the fish populatio stabilizes, the biologist ad the regioal maagers see the commercial possibility of a aual cotrolled harvest. The difficulty would be to maage a sustaiable harvest without depletig the stock. Usig the first model ecoutered i this task with r=.5, determie whether it would be feasible to iitiate a aual harvest of 5 fish after a stable populatio is reached. What would be the ew, stable fish populatio with a aual harvest of this size? From figure. it could be said that the populatio stabilizes at aroud year (due to a roudoff). Most correctly this occurs i year 9 where the populatio strictly is equal to the log-term sustaiable limit. Nevertheless, if oe bega a harvest of 5 fish at ay of these years (depedig o what defiitio of stable is employed), oe could say that u + is reduced by 5, so that whe oe is calculatig the growth of the populatio at the ed of year, oe takes ito accout that 5 fish has bee harvested. Mathematically this is show as: 2 u+ = ( )( u) +.6( u) 5 {2} Table 6.. The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {2}. A harvest of 5 fish is made per aum. (after attaiig stability) Populatio Populatio

14 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 20 years by meas of the logistic fuctio model U+ {2} cosiderig a harvest of 5000 fish per aum Populatio U+ / fish Figure 6.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 20 years usig logistic fuctio model {2}. The model cosiders a aual harvest of 5 fish. The stable fish populatio is 5.0. The behaviour observed i figure 6. resembles a iverse square relatioship, ad although there is o physical asymptote, the curve appears to approach some y-value, util it actually attais the y- coordiate 5.0. It is feasible to iitiate a aual harvest of 5 fish after attaiig a stable populatio (whe that occurs has already bee discussed). From the graph oe ca coclude that the aual stable fish populatio with a aual harvest of 5.0 fish is 5.0x. 7. Ivestigate other harvest sizes. Some aual harvests will cause the populatios to die out. Illustrate your fidigs graphically. a. Cosider a harvest of 7.5. The the logistic fuctio model is: 2 u+ = ( )( u) +.6( u) 7.5 {} Table 7.. The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {}. A harvest of 7.5 fish is made per aum. (after attaiig stability) Populatio Populatio

15 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 20 years by meas of the logistic fuctio model U+ {} cosiderig a harvest of 7500 fish per aum. Populatio U+ / fish Figure 7.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 20 years usig logistic fuctio model {}. The model cosiders a aual harvest of 7.5 fish. The stable fish populatio is.97. From figure 7. oe ca observe this behaviour resemblig a hyperbolic behaviour (x - ). Oe ca coclude that it is feasible to iitiate a aual harvest of 7.5 fish after attaiig a stable populatio 5

16 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese (whe that occurs has already bee discussed). The curve approaches (ad becomes) the value of.22x fish, which is the aual stable fish populatio with a aual harvest of 7.5 fish. b. Cosider a harvest of fish. The the logistic fuctio model is: 2 u+ = ( )( u) +.6( u) {} Table 7.2. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {}. A harvest of x fish is made per aum. (after attaiig stability) Populatio Populatio

17 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 25 years by meas of the logistic fuctio model U+ {} cosiderig a harvest of,000 fish per aum Populatio U+ / fish Figure 7.2. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 25 years usig the logistic fuctio model {}. The model cosiders a aual harvest of x fish. A populatio collapse occurs after year 25. From figure 7.2 oe ca observe the chroic depletio of the fish populatio as a cosequece of excessive harvest. If oe cotiues with the model ito year 26 the populatio turs egative, ad this meas the death of the last fish i the lake. Thus it is ot feasible to harvest above fish per aum, ad oe could impose a prelimiary coditio whe harvestig H fish p.a.: H 8. Fid the maximum aual sustaiable harvest. Whe harvestig ad a stable populatio are attaied, oe could argue that the rate of harvestig equals the rate of growth of the populatio. But sice oe is ot cosiderig a differetial equatio, oe could istead say that the populatio u + modelled i {2}, {} ad {} ca be geeralized with model {5}: 5 2 u ( + = )( u) +.6( u) H {5} Because u =6.0x (a stable value) the r=. Therefore it is valid to write: u = + () u Thus if oe cosidered H=0 oe could write {} as followig 2 u+ = ( )( u+ ) +.6( u+ ) Now, if oe bega harvestig H fish, the equatio would look like the followig: 7

18 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese 2 u+ = ( )( u+ ) +.6( u ) + H 2 ( )( u+ ) +.6( u+ ) u+ H = 0 2 ( )( u+ ) + 0.6( u+ ) H = 0 {6} Model {6} is a quadratic equatio which could be solved, if oe kew the value of the costat term H, yet i this case it serves as a ukow that must be foud. This could be solved by recurrig to a aalysis of the discrimiat of the quadratic equatio: 2 ( )( u+ ) + 0.6( u+ ) H = ± 0.6 ( )( H ) u+ = 2( ) It is kow that there ca be a. Two real roots (D>0, D ) b. Oe real root (D=0, D ) c. No real roots (D<0, D ) For a solutio (exact value for the maximum value for H) oe could cosider a sigle real root: ± 0.6 ( )( H ) 0.6 ± D u+ = = 2( ) 2( ) 2 D = 0.6 ( )( H) = H = = = ( ) Thus, the maximum value of harvest, that still preserves the fish populatio at a stable value is H=9000. This value results i the followig populatio developmet: 2 u+ = ( )( u+ ) +.6( u+ ) 9 {7} Table 8.. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {}. A harvest of x fish is made per aum. (after Populatio Populatio attaiig stability)

19 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese The populatio ultimately reaches the value Estimated magitude of populatio of fish of a hydrolectric project durig the first x^2 (order of magitude) years by meas of the logistic fuctio model U+ {7} cosiderig a harvest of fish per aum (H max) Populatio U+ / fish U+=x^ fish after x^2 (order of magitude) years Figure 8.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of ca. x 2 years (order of magitude) usig the logistic fuctio model {7}. The model cosiders a aual harvest of 9x fish. A stable populatio (x ) is reached after over 0 years (thus the order of magitude). 9. Politicias i the area are axious to show ecoomic beefits from this project ad wish to begi the harvest before the fish populatio reaches its projected steady state. The biologist is called upo to determie how soo fish may be harvested after the iitial itroductio of,000 fish. Agai usig the first model i this task, ivestigate differet iitial populatio sizes from which a harvest of 8,000 fish is sustaiable. The populatio sizes that was ivestigated, was: u 2 This iitial populatio is too small for it to be sustaiable: 9

20 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Table 9.0. The populatio of fish i a lake over a time rage of 20 years estimated usig the logistic fuctio model {}. A harvest of 8x fish is made per aum. (after Populatio Populatio attaiig stability) Thus oe ca say, that the miimum iitial populatio must be u 2 a. Cosider a iitial populatio u =2.5x fish. The equatio could be formulated as follows: 2 u+ = ( )( u+ ) +.6( u+ ) 8 {9} Table 9.. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {}. A harvest of x fish is made per aum. (after attaiig stability) Populatio Populatio

21 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 6 years by meas of the logistic fuctio model U+ {9} cosiderig a harvest of 8000 fish per aum Populatio U+ / fish Figure 9.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 6 years usig the logistic fuctio model {9}. The model cosiders a aual harvest of 8x fish. A stable populatio (x ) is reached. b. Cosider a iitial populatio u =.0x fish. The equatio could be formulated as follows: 2 u+ = ( )( u+ ) +.6( u+ ) 8 {9} Table 9.2. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {9}. A harvest of 8x fish is made per aum. (after attaiig stability) Populatio Populatio

22 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 0 years by meas of the logistic fuctio model U+ {9} cosiderig a harvest of 8000 fish per aum. Iitial populatio is 0,000 fish Populatio U+ / fish Figure 9.2. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 0 years usig the logistic fuctio model {9}. The model cosiders a aual harvest of 8x fish. A stable populatio (x ) is reached. c. Cosider a iitial populatio u =.5x fish. The equatio could be formulated as follows: 2 u+ = ( )( u+ ) +.6( u+ ) 8 {9} Table 9.. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {9}. A harvest of 8x fish is made per aum. (after attaiig stability) Populatio Populatio

23 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 0 years by meas of the logistic fuctio model U+ {9} cosiderig a harvest of 8000 fish per aum. Iitial populatio is 5,000 fish Populatio U+ / fish Figure 9.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 6 years usig the logistic fuctio model {9}. The model cosiders a aual harvest of 8x fish. A stable populatio (x ) is reached. d. Cosider a iitial populatio u =.0x fish. The equatio could be formulated as follows: 2 u+ = ( )( u+ ) +.6( u+ ) 8 {9} Table 9.. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {9}. A harvest of 8x fish is made per aum. (after attaiig stability) Populatio Populatio

24 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 0 years by meas of the logistic fuctio model U+ {9} cosiderig a harvest of 8000 fish per aum. Iitial populatio is 0,000 fish Populatio U+ / fish Figure 9.. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 0 years usig the logistic fuctio model {9}. The model cosiders a aual harvest of 8x fish. A stable populatio (x ) is reached. e. Cosider a iitial populatio u =.5x fish. The equatio could be formulated as follows: 2 u+ = ( )( u+ ) +.6( u+ ) 8 {9} Table 9.5. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {9}. A harvest of 8x fish is made per aum. (after attaiig stability) Populatio Populatio

25 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Estimated magitude of populatio of fish of a hydrolectric project durig the first 0 years by meas of the logistic fuctio model U+ {9} cosiderig a harvest of 8000 fish per aum. Iitial populatio is 5,000 fish Populatio U+ / fish Figure 9.5. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 0 years usig the logistic fuctio model {9}. The model cosiders a aual harvest of 8x fish. A stable populatio (x ) is reached. f. Cosider a iitial populatio u =6.0x fish. The equatio could be formulated as follows: 2 u+ = ( )( u+ ) +.6( u+ ) 8 {9} 25

26 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Table 9.6. The populatio of fish i a lake over a time rage of 0 years estimated usig the logistic fuctio model {9}. A harvest of 8x fish is made per aum. (after attaiig stability) Populatio Populatio Estimated magitude of populatio of fish of a hydrolectric project durig the first 0 years by meas of the logistic fuctio model U+ {9} cosiderig a harvest of 8000 fish per aum. Populatio U+ / fish Figure 9.6. Graphical plot of the fish populatio of the hydroelectric project o a iterval of 0 years usig the logistic fuctio model {9}. The model cosiders a aual harvest of 8x fish. A stable populatio (x ) is reached. Oe ca coclude that the populatio reaches a positive, stable limit i the followig iterval: 26

27 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese u + 5 [2,. [ 5 Also, as u +., the stable populatio istead of tedig (ad ultimately becomig), it 5 teds towards 2, ad whe u + =., the stable populatio is exactly 2. Thus it ca be cocluded, that whe the iitial populatio exceeds the stable limit, it will decrease i magitude, util reachig fish, ad if it lies below the stable populatio limit ( ) it will icrease i magitude util reachig stability. If oe was to plot oe iitial populatio size of u < versus u > a equilibrium (resemblig the chemical equilibrium of a reversible reactio) is attaied: Populatio U+ / fish Estimated magitude of populatio of fish of a hydrolectric project durig the first 0 years by meas of the logistic fuctio model {9} for iitial populatios U =60000 ad U *=0000 with a aual harvest of Figure 9.7. Two iitial populatio sizes attaiig equilibrium at u += for the model {9}. What is the usefuless of such equilibrium? It tells us that if the politicias wated to start such a project with a set aual harvest, they could do this at the lowest cost by startig with a populatio of x fish istead of 6x fish, obtai the same yearly harvest ad keep the populatio alive. This is the major socio-ecoomic beefit of this study. Aother poit to this aalysis is that, if oe started with either of followig values: u = 2 5 u =.2 5 u =. Oe would get to the same stable limit of 2 : u + = ( )(2 ) +.6(2 ) 8 =

28 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese u + = ( )(.2 ) +.6(.2 ) 8 = u + = ( )(. ) +.6(. ) 8 = Whe graphig this oe gets the follow figure: Estimated magitude of populatio of fish of a hydrolectric project durig the first 0 years by meas of the logistic fuctio model {9} for iitial populatios U =20000, U *=20000 ad U **=0000 with a aual harvest of Populatio U+ / fish Figure 9.8. Three iitial populatio sizes developig throughout 0 years time accordig to the model {9}. Thus by startig the project with 2x oe could (theoretically) maitai a stable populatio ad get a fairly high harvest out of it. Coclusio The logistic fuctio {} has bee applied throughout the task ad coclusios ca be made about the limitatios of the model. The model is strictly based o the assumptio that r will remai costat at a value of r=.5 util the stable populatio limit is reached. Thus were the populatio to be iflueced by some exteral factor, the model would ot take this ito accout ad the growth of the populatio would udergo a discrepacy. Oe could argue that the followig assumptios were made whe costructig the model: The fish are ovoviviparous ad carry the egg util it hatches. The fish populatio is somewhat homogeous. Fish reproductio. < (olie). Visited o 2. December

29 IB Mathematics HL Type II Portfolio: Creatig a logistic model Iteratioal School of Helsigborg Christia Jorgese Their rate of reproductio is simplified ad expressed as a growth factor istead of a rate of chage Two cases could occur: a. A exteral factor icreases the mortality of the populatio. b. The fish reproduced at a higher rate tha the model predicts. Also, oe has ot take ito accout that despite the model havig the log-term sustaiability limit at about 6x, several factors could chage this: Huma actio such as cotamiatio of the habitat. Other aquatic vertebrates 2 that might be positioed higher i the food chai, ad could disturb the stability by cosumig the studied fish s ow source of food. Seaso chages Global warmig Thus it would be wise to itroduce a term ito the model that takes ito the accout the log-term sustaiability limit, which i a o-ideal situatio probably would chage. The model predicts a few iterestig thigs: Overpopulatig the lake i the first year leads to the death of all fish by the ed of year oe. The model is built for u =x fish, yet say oe itroduced oe millio fish i the lake (igorig the capacity of the lake): 2 u + = ( )( u) +.6( u) = ( )( ) +.6( ) = 8. fish Populatios that reproduce at a very high rate (ie. r=2.9) result i a dyamic limit (with two values istead of a stable limit with oly oe value) of fluctuatios betwee the extrema, first from the maximum towards the miimum i oly oe year. This opposes the asymptotic behaviour predicted whe the growth factor is smaller. Whe harvestig is itroduced ito a stable populatio (i this case at u =6x ) oly harvests below 9x fish may take place uless extictio of the populatio is desired. Whe the harvestig is set at costat value of 8x, iitial populatios below a specific value will grow i magitude util approachig that value, while iitial populatios above that value (yet ot above a certai limit as overpopulatio causes death as well) lead to a decrease i magitude util approachig the same limit. 2 Biological term for fish: Wikipedia < Visited o 2. December