MATH Differential Equations September 15, 2008 Project 1, Fall 2008 Due: September 24, 2008

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1 MATH 5 - Differenial Equaions Sepember 15, 8 Projec 1, Fall 8 Due: Sepember 4, 8 Lab Logisics Populaion Models wih Harvesing For his projec we consider lab 1.3 of Differenial Equaions pages 146 o 147. Afer reading his maerial consruc a lab repor addressing each of he following quesions for cases 3, 4, 6, 8 of able 1.1 on page 147: 1 1. Given a logisics growh model wih consan harvesing, dp (1 d = kp p ) a, k, N, a R +. (1) N (a) Consruc a lis of variables and parameers associaed wih (3) and describe he meaning of each. (b) Analicall solve (3) using he mehods discussed in secion 1. of he ex. (c) Discuss qualiaive behavior of he soluions o (3) hrough he equaion s: i. Equilibrium Poins ii. Phase Line (d) Using Euler s mehod and a slope field diagram address he following quesion: For a = a 1, wha will happen o he fish populaion for various iniial condiions?. Given a logisics growh model wih periodic harvesing, dp (1 d = kp p ) a(1 + sin(b)), k, N, a, b R +. () N (a) In his case wha do he parameers a and b represen? (b) Is i possible o solve (3) using he mehods discussed in secion 1. of he ex? (c) Using Euler s mehod and a slope field diagram address he following quesions: i. For a = a 1 and b = 1 wha will happen o he fish populaion for various iniial condiions? ii. For a = a and b = 1 wha will happen o he fish populaion for various iniial condiions? Explain wh here are no equilibrium poins and hus no phase line for his problem. 3. Summar and conclusions. In a shor essa forma summarize our resuls from he previous quesions. Compare and conras each of he wo models. Be sure o jusif our conclusions b referencing our previous summar and analsis. 1 Your repor should be well organized and clearl presened. If seps are unclear hen include more seps or make annoaions clarifing he procedure and purpose. Be sure o label and ile an included graphs or ables of daa. 1

2 Lab Repor - Logisics Populaion Models wih Harvesing In he following we respond o quesions associaed wih Lab 1.3 of Differenial Equaions. This repor is organized ino hree secions. In he firs secion we address a logisics growh model equaion wih consan harvesing hrough quaniaive, qualiaive and numerical analses. A similar analsis is conduced on he logisics growh model wih periodic harvesing in he second secion. The hird secion will consis of a summar of resuls and repor of conclusions associaed wih a comparison of hese wo models. 1. Logisics Growh Wih Consan Harvesing (a) Variable and Parameer Lising: dp ( d = f 1(p) = kp 1 p ) a, k, N, a R +. (3) N i. k growh parameer : This parameer describes he rae of growh of he populaion p. ii. N carring capaci : This parameer describes he oal amoun of p ha he resources can suppor. iii. a harvesing rae : This parameer describes he rae ha p will be aken from he ssem. iv. p populaion : This dependen variable describes he populaion as a funcion of ime. v. ime : This independen variable parameerizes he evoluion of he populaion, p. (b) To solve his ODE analicall we noe ha (3) is auonomous and herefore separable. Appling separaion of variables o (3) gives he following: which implies, dp ( kp 1 p n ) a = d = + C, C R (4) = ( A + B ) dp p p 1 p p (5) = A ln p p 1 + B ln p p (6) = ln (p p 1 ) A (p p ) B, (7) (p p 1 ) A (p p ) B = Ce, (8) where p 1 and p are roos o he quadraic polnomial in p and A = (p 1 p ) 1, B = (p p 1 ) 1 are found b parial fracions. If we assume he iniial populaion p() = p is given hen we find ha C = (p p 1 ) A (p p ) B. I is no, in general, clear how we should solve for p explicil. To do his we would need values for N, k, a o find p 1 and p and hus A and B. If hese numbers were known hen polnomial roo finding would give explici formula for p. (c) We now address he qualiaive informaion ha is given b he differenial equaion iself. To do his we firs find he equilibrium soluions of (3) b solving, o ge he equilibrium soluions, dp (1 d = = kp p ) a p Np + an N k p 1 () = p () = N + N N 4 an k N 4 an k =, (9), (1), (11)

3 assuming ha kn 4a. Oherwise, p 1 = p. We also noe ha p is no phsical for N 4 an k > N or N 4 a k < and ha for phsicall relevan cases p 1() > p () for all. To classif hese equilibria we define p 1 = p + and p = p appl linearizaion o ge, ( df dp = k 1 p ) ± (1) p=p± N ( ) = k ± 1 4a, (13) kn which implies ha p 1 = p + is a sink and p = p is a source. Figure 1.1 shows he phase line for he ssem for kn 4a and he special case where p 1 = p. (d) The following able correlaes figures o parameer choices in able 1.1 page 147. Figure Label Figure Tpe Choice Fig. 1. Slope Fields 3 Fig. 1.3 Euler s Mehod 3 Fig. 1.4 Slope Fields 4 Fig. 1.5 Euler s Mehod 4 Fig. 1.6 Slope Fields 6 Fig. 1.7 Euler s Mehod 6 Fig. 1.8 Slope Fields 8 Fig. 1.9 Euler s Mehod 8 From hese figures we can conclude ha as we increase he harvesing parameer he rajecories change giving rise o deca for paricular iniial populaions. This is due o he fac ha as a increases he equilibrium soluions of he ssem ge closer ogeher. As hese equilibria ge closer ogeher he source, p, moves up he p-axis and consequenl decaing rajecories wih iniial populaions beween zero and p are creaed. From his we can also conclude ha as he wo equilibria ge closer and closer o each oher he amoun of rajecories, which grow in ime decreases. Thus i is possible o increase he harvesing parameer o a poin where here are no iniial populaions, which grow in ime. Moreover, i is possible harves o a poin where here are no iniial populaions whose rajecor is viable in he long-erm. Using his model for a biological ssem gives insigh ino how much one could harves wihou desroing he populaion in finie-ime. This can be seen mahemaicall hrough he phase line in figure Logisics Growh Wih Periodic Harvesing dp ( d = f (p, ) = kp 1 p ) a(1 + sin(b)), k, N, a, b R +. (14) N (a) In his case we have ha b is a parameer, which conrols he frequenc of he periodic harvesing and a represens he overall ampliude of he periodic harvesing. We noe ha in his case he harvesing can be as much as a and as lile as. (b) Noing f (p, ) h(p)g() implies ha (14) is no separable. There is no clear wa o solve his differenial equaion analicall. 3

4 (c) Though here are no clear analic soluions we can sill use qualiaive and numerical echniques o he problem. The following able correlaes figures o parameer choices in able 1.1 page 147 for a = a 1 and b = 1. Figure Label Figure Tpe Choice Fig. 1.1 Slope Fields 3 Fig Euler s Mehod 3 Fig. 1.1 Slope Fields 4 Fig Euler s Mehod 4 Fig Slope Fields 6 Fig Euler s Mehod 6 Fig Slope Fields 8 Fig Euler s Mehod 8 Figures 1.18 and 1.19 show slope fields and numerical approximaions for a = a and b = 1. In general we noice similar behavior of he populaion as we did in par (1). Specificall, as we increase he ampliude of he harvesing parameer we creae rajecories are shifed in he negaive direcion. However, in his case, since he ssem is non-auonomous here are no classical equilibrium soluions. Insead, for his ssem, hose rajecories, which do no go exinc end o a sead long-erm oscillaor behavior. 3. Conclusions and Summar In his lab we are presened wo possible models for populaion harvesing. Wih hese models one can conclude ha here is a relaionship beween harvesing rae and he long-erm populaion. We, in general, noe ha increased harvesing leads o more exincion-rajecories. Moreover, i is possible o use he auonomous ssem o infer behaviors of he non-auonomous ssem. While he auonomous ssem has equilibrium soluions he non-auonomous seems o have wha could be considered sead-sae soluions. These sead-sae soluion can be used, like equilibrium soluions, as reference poins o describe he behavior of neighboring soluions. While i is clear how he presence of consan harvesing can shif an auonomous ssem s equilibrium soluion i is no obvious ha his should be rue of a non-auonomous ssem. If we compare Figure 1.1 and Figure 1.6 we see ha for paricular harvesing values boh populaions sele o a rajecor ha remains viable in he long-erm. In conras o his we have Figures 1.8 and 1.16, which show ha he effec of increased harvesing ends o lead o he loss of he populaion in finie-ime. If we hink of hese long-erm paerns as sead-sae soluion hen we can conclude ha he effec of harvesing is o shif hese sead-sae soluions, which creaes iniial populaions desined for exincion similar o he auonomous ssem. Equilibrium soluions as well as sead-sae soluions make i possible o deermine he long-erm behavior of a paricular soluion saisfing an iniial condiion. This is imporan o he phsical reali of he mahemaical model, because one would naurall like o know how much harvesing can ake place wihou decimaing a paricular populaion. Since i is unlikel ha consan harvesing is possible for a paricular populaion i makes sense o consider periodic harvesing insead. While here are clearl qualiaive similariies in he models here are some ineresing quaniaive differences. If we compare Figure 1.13 o Figure 1.5 we see ha oscillaor harvesing has a derimenal effec causing he purple rajecor o go exinc more quickl han in he consan harvesing case. l, sinusoidal harvesing, while more complicaed, does no guaranee he long-erm viabili of all iniial populaions. This is because he harvesing does no ake ino accoun he curren populaion a he ime of harves. A wa o avoid his deca o exincion is o harves he populaion a a rae proporional o he populaion iself. In his wa he harvesing rae will decrease when he populaion is low and possibl migh save a populaion from exincion. This mehod is oulined in problems and 1 of chaper one secion seven of he ex and shows ha his more complicaed auonomous model gives beer(safer) resuls when a populaion ges ver low. 4

5 Figures Figure 1.1 Slope Line for kn 4a. N + N 4 an k p 1 () = N N 4 an k p () =, sink, source Slope Line for kn = 4a. p() = N, node In general, i is no obvious wh his case ields a node. The bes wa o hink abou i is he following: 1. Think of f 1 (p) as a parabola ha opens downward in he f 1 plane and has roos p 1 and p.. As a increases he parabola is shifed in he negaive f 1 direcion. This causes he roos o ge closer ogeher. 3. When kn = 4a he roos join and become a double roo and f 1 is negaive on boh sides of his roo. This implies ha he roo (aka equilibrium soluion) is a node. You ma wan o draw a picure of his o convince ourself. I is ineresing o noe ha his sor of change in he equilibria of an auonomous ODE wih respec o a parameer is called a bifurcaion, specificall a saddle-node bifurcaion, and is highl imporan in he sud of ODE s and heir connecions o mahemaical chaos. We do no have ime for his secion, 1.7 of our ex. I is accessible wih he background we have and hose ineresed should check i ou. 5

6 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.1 max 5 max 1.5 dela = = -.74.**(1-/5)-.1 d/d= -.38 Equaions Runge Kua 4 Draw s max 5 max 1.5 dela.5 6

7 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k)

8 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.16 max 1 max 1.5 dela.5 1 = =.5 1.**(1-/5)-.16 d/d= -.7 Equaions Runge Kua 4 Draw s max 1 max 1.5 dela.5 8

9 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k)

10 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.9 max 1 max 1.5 dela =.718 =.83.**(1-/5)-.9 d/d= -.37 Equaions Runge Kua 4 Draw s max 1 max 1.5 dela.5 1

11 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k)

12 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.4 max 1 max 1.5 dela Equaions Runge Kua 4 Draw s.**(1-/5)-.4 max 1 max 1.5 dela.5 1

13 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k)

14 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.1*(1+sin()) max 5 max 1.5 dela.5 5 = 9.56 =.7 1.**(1-/5)-.1*(1+sin()) d/d= -.9 Equaions Runge Kua 4 Draw s max 5 max 1.5 dela.5 14

15 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) E

16 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.16*(1+sin()) max 1 max 1.5 dela.5 1 = =.5 1.**(1-/5)-.16*(1+sin()) d/d= -.7 Equaions Runge Kua 4 Draw s max 1 max 1.5 dela.5 16

17 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) E

18 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.9*(1+sin()) max 1 max 1.5 dela Equaions Runge Kua 4 Draw s.**(1-/5)-.9*(1+sin()) max 1 max 1.5 dela.5 18

19 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k)

20 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.4*(1+sin()) max 1 max 1.5 dela = = 1.37.**(1-/5)-.4*(1+sin()) d/d= -.54 Equaions Runge Kua 4 Draw s max 1 max 1.5 dela.5

21 Figure DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k)

22 Figure Equaions Runge Kua 4 Draw s.**(1-/5)-.5*(1+sin()) max 1 max 1.5 dela Equaions Runge Kua 4 Draw s.**(1-/5)-.5*(1+sin()) max 1 max 1.5 dela.5 Figure 1.19

23 DELTA T.15 k _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k) _k _k f(_k,_k)