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# Sect9.5 - get/How 3'5_J PAoHe\M 2 X ‘1 Q U...

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Unformatted text preview: get/How 3'5 .____._______J PAoHe \M \ 2 X ‘1: Q U \ gec‘h’m (‘15- gﬁLu-Hms \1 (‘05 1%) ‘— "OSH 1' Xx‘ = GM.“ - z ‘ Y‘Z:—§L, EH: ( ﬁn) wk \giMAMJ SMﬁ’MAF-«kﬂnl a \$144.47 osducuh’m (dean/f“ +\4 UNRVI Section 9.5 Problem 1 parts a and e > with(DEtools): > eqn1:=diff(x(t), t)=x*(1. 5-0. 5*y): eqn2:=diff(y(t), t)=y*(-0. 5+x )= > DEplot([eqn1,eqn2],[x,y],t=0..1,arrows=thin,x=0..1,y=0..6); “NK‘K ///m~n~x\\\\\ / / zh-an\\\\ \ \\ / x f H /M\\\\\\‘1\\\ k \ \\\lw"'lj,‘,1‘,,}; From the direction field it appears that the solutions are all circling the equilibrium at (1/2 3) > ica:=[0,0.2,1],[0,0.2,2],[0,0.2,3],[0,0.2,5]: > icb:=[0,0.4,2.5],[0,0.4,3],[0,0.4,2],[0,0.4,2.25]: > inits:=ica,icb: > DEplot([eqn1,eqn2],[x,y],t=-10..10,{inits},x=0..1,y=0..6,step size=0.025,limitrange=true); QLOHM 3.5 . ; ___ 2 Hum é: '1 X ( l~ 0‘5): ~0f5‘1) = X 0.5x -0.5x~l JFM > ‘3’: (d (—0.15 +0.5)” : —O.ZS} 1- 0.5 x.) AT ) +9» _) @ bF/by 3‘:ij : l—><~‘§,“| 1‘ L: ) ”(r/38c “73“ ’ {‘1 ‘11 + 1)‘ ( ‘ m : (' WARM: (05).! é(‘4 \-Z (lO‘Q/q>(b\‘/)) YR‘t ’3' 00) SW( Mgr—L— At V » r1=‘/q,§”= (A \wwt ‘ Mam a -\ 4 w —. m: (03 mew (1,“ (RR/(3: ( o 3/4 (V) a ‘2‘ ‘4‘; :m; (3‘) sum you‘d "L- ) _. as *hH IL )0) Y r —, (—4 :W LVz 153% a V ) 'J 7' ' gym rmht if?) A“ SM?» (A: *9/31" 1 ﬁm “FM (IE/u. Section 9.5 Problem 3 parts a and e > with(DEtools): > eqn1:=diff(x(t),t)=x*(1-0.5*x-O.5*y):eqn2:=diff(y(t),t)=y*(-0 .25+0.5*x): > DEplot([eqn1,eqn2],[x,y],t=0..1,arrows=thin,x=0..3,y=0..3); V V V V V V From the direction field it appears that the 3 («~N\\\\\\\\\\\\\\\ /«m\\\\\\\\\\\\\\\\ {a-\\\\\\\\\\\\\\\ ‘ I A\\\\\\\\\\\\\\\\ 1/\\\\\\\\\\\\\\\\\ r\\\\\\\\\\\\\\\\\ /\\\\\\\\\\N\\\\\\ /\\\\\\\\\\\\\\\\\ \ ’\ '\ \ \ \ \\\\\\\\\\\ £5 \ \\\\\\\\xxxx~xxx \\ \\\\\\\\\\\\\\\ \\N/ \\\\\\\\\\\\N\ 1 \ \v/ ‘\ \ '\ \\\\\\\\\\ \\wr7i \\\\\\\\K\\N \ \na-d" // ’\ \ \ \N‘x‘x‘x‘a'xm \M-u—r-ﬁx / / '\ \ \\\\\\~\\ 0 5 \u—W/ // / \ \\\\\N‘u\ \W/I / \NKKHH—ov—t «ed-F.) \hJ-x 3 solutions are all spiralling into the equilibrium at (1/2,3/2). ica:=[0,3,0.5],[0,3,1],[0,3,1.5],[0,3,2]: icb:=[0,3,0.1],[0,3,0.2],[O,3,0.3]: icc:=[0,0.5,0.1],[0,0.5,0.2],[0,0.5,0.3],[0,0.5,0.4]: icd:=[0,0.5,0.7],[0,0.5,0.9]: inits:=ica,icb,icc,icd: DEplot([eqn1,eqn2],[x,y],t=-10..20,{inits},x=0..3,y=0..3;step size=0.025,limitrange=true); /Ci.S‘—.o°/ Qwh‘ow ﬁ‘IE/ .. ‘1 ; ‘vvuhlcw 4" X ( M15 -X -0530 : L125»: ~ 11‘ 0‘5 8“] -; F HQ“ X'= ‘3‘: 5(—\+y\ :' ~jf‘x‘1 "— (\(‘h‘ﬂ , '_ q _ _- New (570‘ ft ((1/3 = ( \$4 31)“)3) W: -1451 ) 3‘“: (ﬁrm “WWW SN‘Q - - 1 52A, —‘L+\ﬁ. Writ V7. - 7.1"? ) ‘g - k \ > © A“ Sokw‘ﬁms v; Q‘Yd’jwﬁa‘mt Section 9.5 Problem 4 parts a and e > with(DEtools): , > eqn1:=diff(x(t),t)=x*(1.125-x-0.5*y):eqn2:=diff(y(t),t)=y*(—1 +x): > DEplot([eqn1,eqn2],[x,y],t=0. .1,arrows=thin,x=0..2,y=0..0.5); -—\\ \ \ ‘\ \ \\ \ '\ -—\\\ \ \\ \\ \ \. —\ \\ \ \\ \\ \ '\ —\\\\\\\\\x v-\ \\ \ \\\ N\‘\ -\ \\ \ \\\ \\\ —\ \\\ \\\ \\\ —\ \N ‘\ \\\\\\ \.—\ \\\ \\\\\\ \\\ \\\'\\\\\'\ \-‘\ \\\\\\\\\ \--\ \\\\\\\\\ \\—*\ \\\N\N--~v~. \\\—-\ \‘N‘x‘a‘Ntx‘s‘m \NNW '\ \‘x‘v—o—a-nHw-n \\*-\-—a—— \ \NNNNW \MNHW \ \Hmhmma \\“w 1 Km / {{~\\\\\\\\\x \ \ N H \\ \\ \\ \\ \\ Rx \\\ \\ \\ \ / ///(//drrwwe-—.-—m / / 1’ / ///r(rrae-'W / / ////(r(*///r.p-.-.- / / ///(////Haa(« /////(/u-ﬁ-r--Frv-Fr-u-Fm—v- / //./..-r'e-'-.-' / i m5 1 L5 2 X From the direction field it appears that the solutions are all spiralling into the equilibrium at (1,1/4). ica:=[0, 2,0.1],[0,2,0.2],[0,2,0.3],[0,2,0.4],[0,2;0.05]: icb:=[0,0.25,0.5],[0,0.5,0.5],[0,0.75,0.5]: icc:=[0,0.1,0.1],[0,0.1,0.2],[0,0.1,0.3],[0,0.1,0.4]: icd:=[0,0.5,0.2],[0,0.5,0.15],[0,2,0.15],[0,2,0.25]: inits:=ica,icb,icc,icd: DEplot([eqn1,eqn2],[x,y],t=—10..20,{inits},x=0..2,y=0..0.5,st epsize=0.025,limitrange=true); mSI I \\\\\\ i 0.4 . 03 U 0.2 VVVVVV 0.1 0 ' 0.5 1 1.5 2 gecHM (LS! ‘moMCw 8'- ‘ - ' _ — - ~ ’9 7:1: (a) [ﬁﬁﬁh -—> ’FUJ‘WL‘.‘ - \Jc‘c —7 Ward ‘ m , Fw SwSMU.) (1;, C: 3I so at : 2“— : ﬂ ’ 4 W m V? A! - (q—d . r PML\¢M H ‘H' - X ‘3) Qﬁup‘, 36"“: 15 S. = i at: (-Ct‘lx) X X’j K {£416}- W vaScC'HQ‘A-C’. d __ .. d2; = X (6‘ {2’ ‘73 awn mm Ts 4:1 .. a (444.75») Y = C3 M’ , "6 :1 = U’ ’ wk TM; 1%“. (y) (WW4- ’ WW (3) chum; YMSHWL'A—l 414 efposu'k I» ﬂu, (MW MSM’C, Section 9.5, Problem 12 Problem. Suppose that an insect population :1: is controlled by a natural predator population y according to the model (1), so that there are small cyclic variations in the populations around the critical point (g, g). Suppose that an insecticide is employed With the goal of reducing the population of the insects, and suppose that this insecticide is also toxic to the predators; indeed, suppose that the insecticide kilsl both prey and predator at rates proportional to their respective populations. Write down the modiﬁed differential equations, determine the new equilibrium point, and compare it to the original equilibrium point. To ban insecticides on the basis of this simple model and counterintuitive result would certainly be ill-advised. On the other hand, it is rash to ignore the possible genuine existence of a phenomenon suggested by such a model. Solution. The population model (1) referred to in the problem is the system of differential equations: \$(a - ay) d, y(-C+7w) The problem suggests the pesticide kills a proportion of the population of both predator and prey; note that it does not suggest the same proportion of both are killed. This might turn out to be the case, but for setting up the model one should use the most general problem possible — that the insecticide kills off pct of the insect population and qy of the predator population (Where p and q are constants, and greater than zero). The new model, then, is: x(a — ay) — par dt y(-C+7x) - qy The critical points, Where both if; = 0 and 515:1 = 0, are the points solving both equations: II o Ma—p—aw II o m-C-q+7ﬂ These will be the points (0,0) (total extinction for both populations) and (9151,52). While this has, as one might predict, reduced the population of the v a , predator; it turns out this has increased the population of the insect: %9 > 5' This is the counterintuitive result hinted at in the questiOn’s statement. ...
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Sect9.5 - get/How 3'5_J PAoHe\M 2 X ‘1 Q U...

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