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Unformatted text preview: get/How 3'5 .____._______J PAoHe \M \ 2 X ‘1: Q U \ gec‘h’m (‘15 gﬁLuHms \1 (‘05 1%) ‘— "OSH 1' Xx‘ = GM.“  z ‘
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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. 50. 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\\\\\
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(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
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ﬁm “FM (IE/u. Section 9.5 Problem 3 parts a and e > with(DEtools):
> eqn1:=diff(x(t),t)=x*(10.5*xO.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\\\\\\\\\\\\\\\
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«edF.) \hJx 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 ;
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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
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(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 illadvised. 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 mCq+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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 Spring '04
 Yoon
 Differential Equations, Equations

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