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-1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 280 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 283 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 20 284 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 5 285 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 2 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 286 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 13 287 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 55 "Predator-prey models usi ng the Lotka Volterra equations" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 7 "Made by" }}{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 21 "L10135 Grete Tulinius" }}{PARA 260 "" 0 "" {TEXT -1 13 "Biomodeller G" }}{PARA 266 "" 0 "" {TEXT -1 12 "January 1999" }}{PARA 261 "" 0 "" {TEXT -1 61 "Well-counselled by \+ Jesper J\370rgensen and Jens Christian Larsen" }}{PARA 262 "" 0 "" {TEXT -1 37 "Department of Mathematics and Physics" }}{PARA 263 "" 0 " " {TEXT -1 48 "The Royal Veterinary and Agricultural University" }} {PARA 264 "" 0 "" {TEXT -1 10 "Copenhagen" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Preface" }}{PARA 0 "" 0 "" {TEXT -1 385 "This project was \+ made during a three-week-course in biomodelling at The Royal Veterinar y and Agricultural University in Copenhagen. The aim of the course was to get an insight into many different mathematical models of biologic al systems. The last week and a half of the course was dedicated to gi ving the students a chance to investigate and develop an existing mode l more thoroughly." }}{PARA 0 "" 0 "" {TEXT -1 736 "You will be guided through a number of curves and assumptions of which many are outrageo usly simplified. Biological systems tend to be much more complicated t han they seem at first sight. It is my sincere hope that this report c an contribute to a better understanding of the problem dealt with, whe ther the reader be well-founded in mathematics or not. The mathematics is an important part, and and will be explained as we go along, but i t should be possible to understand the subject without understanding t he details in that respect. Here I must also excuse my miniscule zoolo gical knowledge as of yet - the time limit for this project has made i t impossible to dive too deeply into that part. That could be the aim \+ of future work...." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introducti on" }}{PARA 0 "" 0 "" {TEXT -1 224 "The aim of this project is to anal yse and develop the Lotka Volterra predator-prey model. I have set out to clarify the effects of pesticides or other external forces a syste m of two interacting species in cultivated land. " }}{PARA 0 "" 0 "" {TEXT -1 541 "Pests on crops will here be dealt with as prey for the p redators that to some extent control the pest. The model that we are l ooking into takes its offspring in the model generated by Lotka and Vo lterrra. This model is very simplified indeed and has some weaknesses \+ that will be dealt with as we go along. To this autonomous (time-indep endant) model we will add new parameters and see what happens to the b alance of predator and prey. We will look at seasonal variation and ot her time dependant external forces such as the use of pesticides." }} {PARA 0 "" 0 "" {TEXT -1 382 "Many things must should be taken into ac count here: Do the predator and prey have the same life-cycle? Does th e pesticide diminish the population of predator to the same degree as \+ the prey? What happens if we use species-specific pesticides? Is the t ime span the same for the two to recouperate after a disturbance? Some of these subjects will be left for the reader to dwell upon. " }} {PARA 0 "" 0 "" {TEXT -1 254 "The general thesis that is being tested \+ here is the assumption that in effect using chemical pesticides actual ly enlarges the problem of pests as the cycle of predator-prey around \+ an equilibrium makes a much larger circle after a disturbance than bef ore." }}{PARA 0 "" 0 "" {TEXT -1 362 "I must stress that all conclusio ns can be viewed two ways. It may be the model that is inadequate, so \+ one must must be extremely critical to biological conclusions drawn fr om it. The developing of a model like this should be followed up by te sting its credibility in real-life conditions. This report will only d eal with theoretical assumptions and conclusions. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "The Lotka Volterra Equations" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Introduction" } }{PARA 0 "" 0 "" {TEXT -1 149 "The two scientists Lotka (1925) and Vol terra (1926) independently set up the following model for the populati on dynamics of two interacting species: " }}{PARA 268 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(x(t),t)=rx-axy" "/-%%diffG6$-%\"xG6#%\"t GF),&%#rxG\"\"\"%$axyG!\"\"" }}{PARA 269 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "diff(y(t),t) = -my + bxy" "/-%%diffG6$-%\"yG6#%\"tGF),& %#myG!\"\"%$bxyG\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 330 "r, a, m and b \+ are parameters that we can choose in relation to the species we are de scribing. They tell us something about how fast the species reproduce \+ and the size of the animals compared to eachother. To make things simp le in the beginning we will assume all of these parameters to be one u nless something else is mentioned. " }}{PARA 0 "" 0 "" {TEXT -1 114 " The x represents the population size of the prey and the y represents \+ the predator. From this we can see that the " }{TEXT 256 6 "change" } {TEXT -1 150 " in the population of prey is dependant upon the size of the population of the prey itself subtracting a multiple of the numbe r of prey and predators." }}{PARA 0 "" 0 "" {TEXT -1 201 "The logic of the model is quite obvious, really: A large population of prey will e nhance the number of predators whereas on the other hand a large popul ation of predators will limit the number of prey." }}{PARA 0 "" 0 "" {TEXT -1 300 "And since the two equations are interdependent, the rise and fall in number of prey and predators will go on only to a certain extent. At some point the population of predators will be so big and \+ the population of prey so small that the predators start dying. And fr om here the circle goes on and on." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Analysing the model in detail" } }{SECT 1 {PARA 278 "" 0 "" {TEXT -1 39 "What happens if there are no p redators?" }}{PARA 0 "" 0 "" {TEXT -1 86 "Let us try to analyse the eq uations further before we start solving and plotting them." }}{PARA 0 "" 0 "" {TEXT -1 110 "There are no predators only prey present on the \+ field. To make things simple we set the parameters r= m=a=b=1." }} {PARA 0 "" 0 "" {TEXT -1 21 " This gives us: " }{XPPEDIT 18 0 "dx /dt=x" "/*&%#dxG\"\"\"%#dtG!\"\"%\"xG" }{TEXT -1 233 " , which shows \+ us that Lotka Volterra makes an assumption that for a number of prey t he growth will be that same number, giving us unlimited exponential gr owth. Differently phrased the solution to the above differential equat ion is:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "restart;\neq:=\{d iff(x(t),t)=x\}:\ninit:= \{x(0)=1\}:\nsol:= dsolve(eq union init,x(t)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "solx:= subs(sol,x(t)): with(DEtools):\nf:=unapply(solx,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"fG%$expG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(f,0.. 5,title=`No predators`);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7 U7$\"\"!$\"\"\"F(7$$\"1LLL3x&)*3\"!#;$\"1(RBE[Y^6\"!#:7$$\"1nm\"H2P\"Q ?F.$\"1!)*HVspgA\"F17$$\"1LL$eRwX5$F.$\"1x)GU@\\SO\"F17$$\"1ML$3x%3yTF .$\"1+09%zH'=:F17$$\"1nm\"z%4\\Y_F.$\"1B%H;wl)*o\"F17$$\"1MLeR-/PiF.$ \"10QmBj#e'=F17$$\"1***\\il'pisF.$\"1_#**HGat1#F17$$\"1MLe*)>VB$)F.$\" 1*R%4V()p)H#F17$$\"1++DJbw!Q*F.$\"1K.bi@1bDF17$$\"1nm;/j$o/\"F1$\"1#QK ^mC'[GF17$$\"1LL3_>jU6F1$\"1Os.4(3]8$F17$$\"1++]i^Z]7F1$\"1aNjJ=+#\\$F 17$$\"1++](=h(e8F1$\"1(Qbsjp8*QF17$$\"1++]P[6j9F1$\"1_-$H!GR>VF17$$\"1 L$e*[z(yb\"F1$\"1;zq!\\L([ZF17$$\"1nm;a/cq;F1$\"1LD@ee9:`F17$$\"1nmm;t ,m$\\36=v%eF17$$\"1+]iSj0x=F1$\"1v@\"=%>CMlF17$$\"1nmm\"pW`(>F 1$\"1gosVT54sF17$$\"1+]i!f#=$3#F1$\"1+dAVW)*H!)F17$$\"1+](=xpe=#F1$\"1 j/J`ZQ)*))F17$$\"1nm\"H28IH#F1$\"13[&)ekt/**F17$$\"1n;zpSS\"R#F1$\"1\" =u7!G)G4\"!#97$$\"1LL3_?`(\\#F1$\"1>x*p_Y_@\"Fjr7$$\"1M$e*)>pxg#F1$\"1 (R`()yuoN\"Fjr7$$\"1+]Pf4t.FF1$\"1iH<4^`$\\\"Fjr7$$\"1MLe*Gst!GF1$\"1r k!e%Hjc;Fjr7$$\"1+++DRW9HF1$\"1$oAJjbQ%=Fjr7$$\"1++DJE>>IF1$\"1#)3CLaZ Z?Fjr7$$\"1+]i!RU07$F1$\"1KG*ehmeE#Fjr7$$\"1++v=S2LKF1$\"1Z!=um[d`#Fjr 7$$\"1mmm\"p)=MLF1$\"1?+fB7c0GFjr7$$\"1++](=]@W$F1$\"15b57)3a7$Fjr7$$ \"1L$e*[$z*RNF1$\"1ZA/U2iYMFjr7$$\"1,+]iC$pk$F1$\"1CnBSCoNQFjr7$$\"1m; H2qcZPF1$\"1y(QmUvX'Fjr7$$\"1ML3xe,tUF1$\"1x;8nhwtrFjr7$$\"1n;HdO=yVF1$ \"1^BCz^JpzFjr7$$\"1,++D>#[Z%F1$\"1g=#yx'*yx)Fjr7$$\"1nmT&G!e&e%F1$\"1 +]fYt+1)*Fjr7$$\"1MLL$)Qk%o%F1$\"1J8#33 " 0 "" {MPLTEXT 1 0 79 "eq2:=\{diff(y(t),t)=-y\}:\ninit2:= \{y(0)=10\}:\ns ol2:= dsolve(eq2 union init2,y(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "soly:= subs(sol2,y(t)):\ng:=unapply(soly,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6#%\"tG6\"6$%)operatorG%&arrowG F(,$*$-%$expG6#9$!\"\"\"#5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(g,0..10,title=`No prey`);" }}{PARA 13 "" 1 "" {INLPLOT "6 %-%'CURVESG6$7Z7$\"\"!$\"#5F(7$$\"1mmmT&)G\\a!#<$\"1c,(4Y_'p%*!#:7$$\" 1LLL3x&)*3\"!#;$\"1=e:t'p?RvRB F17$$\"1nm\"zR'ok;F1$\"1Caqy**\\#*=F17$$\"1++D1J:w=F1$\"1'Q&*=M#yJ:F17 $$\"1MLL3En$4#F1$\"1DvSXrLK7F17$$\"1nm;/RE&G#F1$\"1;(402su,\"F17$$\"1+ ++D.&4]#F1$\"1PY?DGq+#)F57$$\"1+++vB_%ResRNF57$$\"1MLLLY.KNF1$\"1NIXs^`CHF57$$\"1++D\"o7Tv$F1$\"157D[ B7UBF57$$\"1LLL$Q*o]RF1$\"1+O#zCVT#>F57$$\"1,+D\"=lj;%F1$\"1yT%*[G&3b \"F57$$\"1++vV&R5F5 7$$\"1MLeR\"3Gy%F1$\"1N+V.\\Xs$)F.7$$\"1nm;/T1&*\\F1$\"1VLMY2aF1$\"1]0b7S+$[%F.7$$\"1nm; zXu9cF1$\"1o#fkESPk$F.7$$\"1+++]y))GeF1$\"1GF.7$$\"1++]P![hY'F1$\"11 O`+[?b:F.7$$\"1LLL$Qx$omF1$\"1I!Qmae/F\"F.7$$\"1+++v.I%)oF1$\"1zIYL@tB 5F.7$$\"1mm\"zpe*zqF1$\"1gD\"3;z!=%)!#=7$$\"1,++D\\'QH(F1$\"1]:Nc*fpz' Fex7$$\"1LLe9S8&\\(F1$\"1'>_Fex7$$\"1NLe 9tOc()F1$\"1\\7Fubbu:Fex7$$\"1,++]Qk\\*)F1$\"1[RkTP$yH\"Fex7$$\"1NL$3d g6<*F1$\"15&zBXd*R5Fex7$$\"1ommmxGp$*F1$\"1t)oGK7/`)!#>7$$\"1++D\"oK0e *F1$\"1Zn\"z3:g!pF]\\l7$$\"1,+v=5s#y*F1$\"1&H#pOs!=k&F]\\l7$F)$\"1%[[i (H**RXF]\\l-%'COLOURG6&%$RGBG$F*!\"\"F(F(-%&TITLEG6#%(No~preyG-%%VIEWG 6$;F(F)%(DEFAULTG" 2 225 218 218 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 456 14300 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 153 "For the development in number of predators with no prey it could apply - if t he sole source of food for the predator is realistically the prey ment ioned." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 "To find the steady state where the two species are in balance (wh ere growth=decline=0) we find the point(s) where both equations=0, the so-called singular points for the system." }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{SECT 1 {PARA 275 "" 0 "" {TEXT 266 20 "Finding equili briums" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "LotkaVolterra := [diff(x(t),t) = r*x - a*x*y, diff(y( t),t) = -m*y + b*x*y];r:=1:a:=1:m:=1:b:=1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.LotkaVolterraG7$/-%%diffG6$-%\"xG6#%\"tGF-,&*&%\"rG \"\"\"F+F1F1*(%\"aGF1F+F1%\"yGF1!\"\"/-F(6$-F4F,F-,&*&%\"mGF1F4F1F5*(% \"bGF1F+F1F4F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "rightha ndsides:=map(rhs,LotkaVolterra):\nsingpoints:=solve(\{op(righthandside s)\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+singpointsG6$<$/% \"xG\"\"!/%\"yGF)<$/F+\"\"\"/F(F." }}}{PARA 0 "" 0 "" {TEXT -1 79 "So \+ there are two equilibriums to the system, namely (x,y)=(0,0) and (x,y) =(1,1)" }}{PARA 0 "" 0 "" {TEXT -1 89 "To analyse these steady states \+ further we must find the Jacobians to the singular points:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "with (linalg):A:=jacobian(righthandsides,[x,y]):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "In the points (0,0) and (1,1) the Jacobians are:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "A1:=subs(singpoints[1],evalm (A));A2:=subs(singpoints[2],evalm(A));" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'MATRIXG6#7$7$\"\"\"\"\"! 7$F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'MATRIXG6#7$7$\" \"!!\"\"7$\"\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "Then we fi nd the eigenvectors, trace, determinant and discriminant for both poin ts, so we can determine how the functions behave in the vicinity of th e singular points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "eige nvectors(A1);sigma1:=trace(A1);delta1:=det(A1);disk:=(s,d)->s^2-4*d:De lta1:=disk(sigma1,delta1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%!\"\" \"\"\"<#-%'VECTORG6#7$\"\"!F%7%F%F%<#-F(6#7$F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'d elta1G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Delta1G\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "eigenvectors(A2);sigma2:=tra ce(A2);delta2:=det(A2);Delta2:=disk(sigma2,delta2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$7%%\"IG\"\"\"<#-%'VECTORG6#7$F%,$F$!\"\"7%F+F%<#-F(6# 7$F%F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigma2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'delta2G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Delta2G!\"%" }}}{PARA 5 "" 0 "" {TEXT 268 38 "In (0,0) we hav e a saddlepoint (since " }{XPPEDIT 270 0 "Delta>0" "2\"\"!%&DeltaG" } {TEXT 269 5 " and " }{XPPEDIT 272 0 "delta<0" "2%&deltaG\"\"!" }{TEXT 271 119 " ), which means that unless both populations are excactly 0 f rom the beginning we will not see that situation occur. " }}{PARA 0 "" 0 "" {TEXT -1 103 "The second equlibrium is much more interesting. \+ Because we have only complex eigenvalues and the trace " }{XPPEDIT 18 0 "sigma=0" "/%&sigmaG\"\"!" }{TEXT -1 258 ", we can deduct that (x,y) = (1,1) is a center around which the the model circles. This means tha t depending on the initial conditions the function of x and y will cir cle around (1,1), but never actually reach that point. The initial con ditions detemine only " }{TEXT 273 7 "how far" }{TEXT -1 33 " from (1, 1) the functions circle." }}}{PARA 0 "" 0 "" {TEXT -1 37 "We move on t o graphic demonstrations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 14 "Phaseportraits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots):with(D Etools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "init1 := x(0)=1 .5, y(0)=0.5:init2 := x(0)=0.8,y(0)=.8:init3:= x(0)=0,y(0)=0:init4:= x (0)=1,y(0)=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "P1:=phase portrait(LotkaVolterra,[x(t),y(t)],0..10,[[init1],[init2],[init3],[ini t4]],stepsize=.1,linecolour=[green,blue,coral,red], title=`Phaseportra it 1`):P1;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6*-%'CURVESG6]dl7%7$\"\"!F(F'F'7%7$$\"+NWfg^!#6F(7$$\"+J$y\" [:!#5F(7$$\"+P!=ZK\"F1$\"+x&*y*G\"F-7%7$$\"+I$y\"[:F1F(7$$\"+=sH!e#F1F (7$$\"+Cp$oN#F1$\"+y&*y*G\"F-7%7$$\"+`aF1FW7%7$$\"+yQlwcF1F(7 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]^lFi[dlFjc[nF^d[n-%%VIEWG6$;$!+)QCAH&F*$\"+'\\`>6\"F]^l;$!+'p6\"F]^l-%+AXESLABELSG6$%\"xG%\"yG-%&TITLEG6#%0Phaseportrait~2G " 2 258 265 265 6 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10010 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 79 158 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 238 "Here the smallest circle \+ (which was the largest in the previous phaseportrait) is the function \+ before disturbance, the middle circle is with drastic reduction of pre y only, and the largest circle is with drastic reduction of both speci es." }}{PARA 0 "" 0 "" {TEXT -1 84 "As is easily seen the farmer may s olve his immediate problem and reduce the pest to " }{TEXT 261 6 "almo st" }{TEXT -1 136 " zero, but in the long run, what he has really done , is disturb the balance and enlarge his own problem. (All according t o this model.) " }}{PARA 0 "" 0 "" {TEXT -1 195 "However any disturban ce, manmade or not, can cause the model to behave like this. So in the case of a harsh winter or a very small invasion of both predator and \+ prey we will get the same results." }}{PARA 0 "" 0 "" {TEXT -1 123 "Th is also means that reduction (if it is controllable) to only a certain degree (close to the steady state) will give us a " }{TEXT 267 6 "bet ter" }{TEXT -1 9 " balance." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "So lutions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Let us move on to look ing at the solutions to the Lotka Volterra equations. Solutions cannot be found analytically so we will let Maple solve them numerically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 11 "Soluti on #1" }}{PARA 0 "" 0 "" {TEXT -1 59 "The first one is for initial con ditions x(0)=1.5, y(0)=0.5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "LotkaVolterra1 := \{op(LotkaVolterra)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/LotkaVolterra1G<$/-%%diffG6$-%\"yG6#%\"tGF-,&F+!\"\" *&%\"xG\"\"\"F+F2F2/-F(6$-F1F,F-,&F1F2F0F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 92 "LVsol1:=dsolve(LotkaVolterra1 union \{init1\},[x(t) ,y(t)],\ntype=numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "xsol1:=subs(LVsol1,x(t)): ysol1:=subs(LVsol1,y(t )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([xsol1,ysol1],0 ..40,0..11,title=`Prey=1.5 Predator=0.5`);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7iz7$\"\"!$\"#:!\"\"7$$\"+4x&)*3\"!#5$\"+`2[\" e\"!\"*7$$\"+F27$$\"+m;')=()F/$\"+C%[6'>F27$ $\"+R?ah5F2$\"+.06I>F27$$\"+7z>^7F2$\"+LbHE=F27$$\"+\\e-Y8F2$\"++6:\\< 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The overall picture is almost the same, so we will not dwell on t hese graphs - look at them and draw your own conclusions." }}{PARA 271 "" 0 "" {TEXT -1 1 " " }}{PARA 5 "" 0 "" {TEXT 259 255 "If we set \+ the birthrate of the prey (r) to 0.5 and the deathrate of the predator (m) to 0.5 while we set the interactionparameters (a and b) to 1 and \+ 0.5 (on the assumption that one predator needs two prey to stay alive) we get not so different solutions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 274 11 "Solution #1" }}{PARA 0 "" 0 "" {TEXT -1 64 "The first graph is for initial conditions x(0)=1.5 and y(0)=0.5 :" }}{PARA 5 "" 0 "" {TEXT 260 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "r:=0.5:m:=0.5:a:=1:b:=0.5:LotkaVolterra := [diff(x(t ),t) = r*x - a*x*y, diff(y(t),t) = -m*y + b*x*y];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.LotkaVolterraG7$/-%%diffG6$-%\"xG6#%\"tGF-,&F+$\"\"& !\"\"*&F+\"\"\"%\"yGF3F1/-F(6$-F4F,F-,&F4$!\"&F1F2F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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parameters" }}{PARA 272 "" 0 "" {TEXT -1 125 "We can also try a graph showing the interrelation if the deathrat e of the predator is smaller than the birthrate of the prey." }}{PARA 272 "" 0 "" {TEXT -1 28 "So r=0.5, m=.3, a=1, b=0.5. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r:=.5:m:=.2:a:=1:b:=.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "LotkaVolterra := [diff(x(t),t) = r* x - a*x*y, diff(y(t),t) = -m*y + b*x*y];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.LotkaVolterraG7$/-%%diffG6$-%\"xG6#%\"tGF-,&F+$\"\"&!\"\"*&F+ \"\"\"%\"yGF3F1/-F(6$-F4F,F-,&F4$!\"#F1F2F/" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 277 9 "Solution " }}{PARA 0 "" 0 "" {TEXT -1 56 "Again we try \+ with initial conditions x(0)=1.5, y(0)=0.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "LotkaVolterra1 := \{op(LotkaVolterra)\}:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "LVsol1:=dsolve(LotkaVolterra 1 union \{init1\},[x(t),y(t)],\ntype=numeric,output=listprocedure):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "xsol1:=subs(LVsol1,x(t)): \+ ysol1:=subs(LVsol1,y(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot([xsol1,ysol1],0..60,0..11, title=`Prey=1.5 Predator=0.5`);" } }{PARA 273 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CUR VESG6$7\\s7$\"\"!$\"#:!\"\"7$$\"+DJdpK!#5$\"+Np$oZ\"!\"*7$$\"+]i9RlF/$ \"+;$4]S\"F27$$\"+8G$R<)F/$\"+TZF^8F27$$\"+v$>(3)*F/$\"+ep)pG\"F27$$\" +%f]V9\"F2$\"+\")e!Q@\"F27$$\"+]#HyI\"F2$\"+>q!Q8\"F27$$\"+b62]9F2$\"+ eaYg5F27$$\"+gIJ#f\"F2$\"+Wo6`)*F/7$$\"+l\\bM>RF gq7$$\"+)Q=\"))**F2$\"+b(H<,%Fgq7$$\"+k=pD6!\")$\"+H\"4?l%Fgq7$$\"+lN? 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So next step is to modify and see what \+ happens!" }}}}}{PARA 0 "" 0 "" {TEXT -1 173 "So far we have seen how t he original Lotka Volterra equations behave. But as stated earlier the biological credibility is not very high making the interpretations un certain." }}{PARA 0 "" 0 "" {TEXT -1 34 "Maybe we can improve the mode l...." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 "Modifications to the Lotka Volterra Equations" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 278 17 "Subtracting p(t)x" }}{PARA 0 "" 0 "" {TEXT -1 132 "We shall add a time-dependent parameter to the Lotka Vol terra equations. This parameter can represent the amount of pesticide \+ added." }}{PARA 280 "" 0 "" {XPPEDIT 18 0 "dx/dt=rx-axy-p(t)*x" "/*&%# dxG\"\"\"%#dtG!\"\",(%#rxGF%%$axyGF'*&-%\"pG6#%\"tGF%%\"xGF%F'" } {MPLTEXT 1 0 3 " " }{XPPEDIT 18 0 "dy/dt=-my+ bxy " "/*&%#dyG\"\" \"%#dtG!\"\",&%#myGF'%$bxyGF%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "Here whatever the disturbance is, only th e prey is affected by it." }}{PARA 0 "" 0 "" {TEXT -1 146 "Setting p t o be constant would not be very interesting. That has the same effect \+ as changing the birthrate of the prey, so we will skip that part." }} {SECT 0 {PARA 4 "" 0 "" {TEXT 279 20 "Finding equilibriums" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "r:=1:m:=1:a:=1:b:=1:modLotkaVolterra:=[diff(x(t),t) =r*x-a*x*y-p(t)*x,diff(y(t),t)=-m*y+b*x*y];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1modLotkaVolterraG7$/-%%diffG6$-%\"xG6#%\"tGF-,(F+\" \"\"*&F+F/%\"yGF/!\"\"*&-%\"pGF,F/F+F/F2/-F(6$-F1F,F-,&F1F2F0F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "modrighthandsides:=map(rhs,m odLotkaVolterra):\nmodsingpoints:=solve(\{op(modrighthandsides)\},\{x, y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.modsingpointsG6$<$/%\"xG\" \"!/%\"yGF)<$/F(\"\"\"/F+,&F.F.-%\"pG6#%\"tG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "Again we have two singular points (x,y)=(0,0) and (x,y)=(1,1-p(t)). This means that any value for p(t)>1 will have no biological meaning. We will not analyse (x,y)=(0,0), since it is uninteresting in this context." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "with(linalg):modA:=jacobian(modrigh thandsides,[x,y]):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new defini tion for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition \+ for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "modA2:=subs(mo dsingpoints[2],evalm(modA));" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&modA2G-%'MATRIXG6#7$7$\"\"!!\"\"7$, &\"\"\"F.-%\"pG6#%\"tGF+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Aga in we find eigenvalues, trace, determinant and discriminant for (1,1-p (t))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eigenvalues(modA2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$,&!\"\"\"\"\"-%\"pG6#%\"tGF&#F& \"\"#,$F#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "disk:=(s,d)- >s^2-4*d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "modsigma2:=tra ce(modA2);moddelta2:=det(modA2);modDelta2:=disk(modsigma2,moddelta2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*modsigma2G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*moddelta2G,&\"\"\"F&-%\"pG6#%\"tG!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*modDelta2G,&!\"%\"\"\"-%\"pG6#%\"tG \"\"%" }}}{PARA 285 "" 0 "" {TEXT -1 144 "Assuming that p(t)<1 we will get a negative discriminant and the trace=0, giving us a center just \+ like in the original Lotka Volterra equations." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 280 13 "Defining p(t)" }}{EXCHG {PARA 286 "" 0 "" {TEXT -1 380 "Now let us try to define the function p(t). We need something tha t expresses how many survivors we will have after a spraying. P(t) is \+ a combination of two functions. The effect of the pesticide must decli ne exponentially - it cannot be a constant effect since the pesticide \+ will eventually be washed away or be decomposed chemically. So for the pesticide we choose the function " }{XPPEDIT 18 0 "K(t)=K0*exp(-qt) " "/-%\"KG6#%\"tG*&%#K0G\"\"\"-%$expG6#,$%#qtG!\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 97 "Here K0 is the dosage at application and q is a parame ter for the deterioration of the pesticide." }}{PARA 0 "" 0 "" {TEXT -1 22 "The other function is " }{XPPEDIT 18 0 "f(t)=B*(1-1/(1+(K/C)^d) " "/-%\"fG6#%\"tG*&%\"BG\"\"\",&F)F)*&F)F),&F)F))*&%\"KGF)%\"CG!\"\"% \"dGF)F1F1F)" }{TEXT -1 438 ". As a whole this expresses the effective ness of the pesticide. The parameter B determines the maximum effectiv eness of the pesticide, so that even with high K(t) the effectiveness \+ can get no higher. The C is the LD50 dose for the pesticide - meaning \+ the dose at which 50% of tested population died in laboratory-experime nts. The parameter d expresses the prey's susceptibility to the pestic ide. The higher the d, the greater the effect. " }}{PARA 0 "" 0 "" {TEXT -1 63 "So p(t) is a combination of the two functions - namely f( K(t))." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f:=k->B*(1-1/(1+( k/C)^d));\nK:=t->K0*exp(-q*t);\np:=t->f(K(t)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"kG6\"6$%)operatorG%&arrowGF(*&%\"BG\"\"\",& F.F.*$,&F.F.)*&9$F.%\"CG!\"\"%\"dGF.F6F6F.F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG:6#%\"tG6\"6$%)operatorG%&arrowGF(*&%#K0G\"\"\"-% $expG6#,$*&%\"qGF.9$F.!\"\"F.F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 281 14 "Phaseportraits" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "To start with w e set the parameters to B=0.1, C=1, d=3, q=1, K0=2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B:=.1:C:=1:d:=3:q:=1:K0:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots):with(DEtools):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "init1 := x(0)=.9, y(0)=2:ini t2 := x(0)=1,y(0)=.2:init3:= x(0)=2,y(0)=.5:init4:= x(0)=1,y(0)=1:" }} 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Only for the initial conditi on x(0)=2, y(0)=0.5 (less predators than prey) it has a positive effec t with time." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Then we change t he parameters B and K0 to be: B=1 and K0=3, thus strengthening the eff ectivity of the pesticide." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "B:=1:K0:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "phasepo rtrait(modLotkaVolterra,[x(t),y(t)],0..30,[[init1],[init2],[init3],[in it4]],stepsize=.1,linecolour=[blue,coral,red,green],title=`B=1, K0=3`) ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6)-%'C URVESG6[^l7$7$$\"\"\"\"\"!$\"\"#!\"\"7$$\"1 F17$F.7$$\"1(ps)*))G9q*F1$\"1`md'4CQ*>F17$F57$$\"1-(QckU)z&*F1$\"1F17$F;7$$\"1/B2okR#[*F1$\"11mMC()Hx>F17$FA7$$\"1!Hs-m#[9%*F1$\"1a z9)Qpj'>F17$FG7$$\"1\"*G^Fp\\#Q*F1$\"1.'zZq7X&>F17$FM7$$\"1mR%=y@QR*F1 $\"1\"Q*\\_T^U>F17$FS7$$\"1^0'RfzmX*F1$\"1!Gf=s\"HJ>F17$FY7$$\"1\"f\"3 h^#)z&*F1$\"1Nh\"p^1>#>F17$Fin7$$\"1jUvNH3s(*F1$\"1IMoD;d:>F17$F_o7$$ 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IXYgv5Fho7$Fe\\fl7$$\"1D/@M7D6#*F1$\"1n))yw.]n5Fho7$F[]fl7$$\"1&Hy9\"*F17$Fgcfl7$$\"1fL?&)>)pd*F1$\"1wO8A0rw!*F17$F]d fl7$$\"1)\\$)*\\t`n'*F1$\"1*)y&pedC/*F17$Fcdfl7$$\"1?Y7/6)=w*F1$\"1U3B 'Qrm,*F17$Fidfl7$$\"12pw!Rg#f)*F1$\"1E%[ob(e***)F17$F_efl7$$\"1b/yG5() e**F1$\"1z+E4SR\"**)F17$Feefl7$$\"1F,A7f)f+\"Fho$\"1(\\F[6HA**)F17$F[f fl7$$\"1Tui\"QKh,\"Fho$\"1O+\"*z\"z@+*F17$Faffl7$$\"1=n:S@BE5Fho$\"1,- %=?v7-*F17$Fgffl7$$\"1M/Q&4(=O5Fho$\"1j_@M*)[\\!*F17$F]gfl7$$\"1,xBnh* e/\"Fho$\"1X#ymKGn3*F17$Fcgfl7$$\"1>nG/iDb5Fho$\"1oj&yWKG8*F17$Figfl7$ $\"12En*3jT1\"Fho$\"1qn5\\gc(=*F17$F_hfl7$$\"1R%Rt.8D2\"Fho$\"1AN@!)Rh ]#*F17$Fehfl7$$\"1D(e12/-3\"Fho$\"1QVJCdd@$*F17$F[ifl7$$\"1Yb%emPr3\"F ho$\"1n*)=v.'**R*F17$Faifl7$$\"1vOeJ0A$4\"Fho$\"1f'o:V#=&[*F17$Fgifl7$ $\"1-3n%[m$)4\"Fho$\"1Z1?D)eld*F17$F]jfl7$$\"1:\\\"*p$)\\-6Fho$\"1Hi-E wIt'*F17$Fcjfl7$$\"1\"**R='*\\b5\"Fho$\"1Wobs)[Xx*F17$Fijfl7$$\"1[5,xx Y26Fho$\"1*[(4_zIz)*F17$F_[gl7$$\"1FO/?F@36Fho$\"1F:k)*=_')**F17$Fe[gl 7$$\"1Duk%[hx5\"Fho$\"1CFG))[]45Fho7$F[\\gl7$$\"1Bm?Mw516Fho$\"1z*3Q5o .-\"Fho7$Fa\\gl7$$\"1;p#\\HiK5\"Fho$\"1_)o.1;6.\"Fho7$Fg\\gl7$$\"1Tl_3 WD*4\"Fho$\"1o&ew\"*>;/\"Fho7$F]]gl7$$\"1b2J508%4\"Fho$\"106Cm*\\<0\"F ho7$Fc]gl7$$\"1Ws;@S&z3\"Fho$\"1??nJ!y81\"Fho7$Fi]gl7$$\"1!zdp1/33\"Fh o$\"12B$*3+Qq5Fho7$F_^gl7$$\"1H\"))R'Qxs5Fho$\"1CVYM%Q'y5Fho7$Fe^gl7$$ \"1B1eK(oR1\"Fho$\"1]CbH;]W5Fho$\"1hal(zGz4\"Fho-Fb\\r6&Fd\\rF*Fe\\rF* Fh\\rF\\]r-%%VIEWG6$;$\"*Hww9\"!#5$\"+nht#G$!\"*;$\"*(*)e[6Fd`gl$\"+Fe Y#G$Fg`gl-%+AXESLABELSG6$%\"xG%\"yG-%&TITLEG6#%*B=1,~K0=3G" 2 282 240 240 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 180 192 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "We see the same pattern." }} {PARA 0 "" 0 "" {TEXT -1 95 "One last try. See what happens if we set \+ B=1 and K0=10 - a very large dosis of pesticide added:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "B:=1:K0:=10:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "phaseportrait(modLotkaVolterra,[x(t),y(t)],0..30,[[i nit1],[init2],[init3],[init4]],stepsize=.1,linecolour=[blue,coral,red, green],title=`B=1, K0=10`);" }}{PARA 13 "" 1 "" {INLPLOT "6)-%'CURVESG 6[^l7$7$$\"\"\"\"\"!F(7$$\"1_Sb$R'*30*!#;$\"169L`2$=&**F.7$F+7$$\"1QKG QL')*>)F.$\"1`zw\\*y^\")*F.7$F27$$\"1HYK'f8BW(F.$\"1*[]vw-Hg*F.7$F87$$ \"1!\\aAZ'yrnF.$\"1i(**oKE%G$*F.7$F>7$$\"1!Rzv,03='F.$\"1xa?Yc([+*F.7$ FD7$$\"1B%*=\\*Q:m&F.$\"1<;!4(H[W')F.7$FJ7$$\"1aN0e*yi?&F.$\"1G)R-iG#e #)F.7$FP7$$\"17e7<*\\x![F.$\"1fAgj=nbyF.7$FV7$$\"1mql>&)HfWF.$\"1aG%p[ >\\W(F.7$Ffn7$$\"1A*f*>.*\\:%F.$\"1#RP%[ioKqF.7$F\\o7$$\"1ucwsik*)QF.$ \"1O>vy/TCmF.7$Fbo7$$\"1K@)G]U)eOF.$\"1$G6Wf(RCiF.7$Fho7$$\"1R!Q$G2*)e MF.$\"1f%y=osf$eF.7$F^p7$$\"1nyZE6\"oG$F.$\"1g'oS)*H;Y&F.7$Fdp7$$\"1u8 ^?XISJF.$\"1.%z.0sJ5&F.7$Fjp7$$\"1=jCSjs11=HF.$\"1I()f3kRQWF.7$Ffq7$$\"1[4NK$))3%GF.$\"1K'HcErK8%F.7$F\\r 7$$\"1e/BY7M'y#F.$\"1_B!)fqeYQF.7$Fbr7$$\"1Vd()HF.$\"1nV()*RA#)\\#F.7$Fft7$$\"1Uuk[ [n?JF.$\"1oh5$>G0L#F.7$F\\u7$$\"1NzTt0;&G$F.$\"1G#R%[hKx@F.7$Fbu7$$\"1 h7Ln!y>[$F.$\"1`.8E;(y.#F.7$Fhu7$$\"1!z(fx5;7PF.$\"13mbMPU6>F.7$F^v7$$ \"1kGo(RGp(RF.$\"1J$>K3isz\"F.7$Fdv7$$\"1.]F=iwxUF.$\"1X<**GCq%p\"F.7$ Fjv7$$\"1)>W&pw[;YF.$\"13jlj36.;F.7$F`w7$$\"1UL&>Cr_*\\F.$\"1<^yzl\">_ \"F.7$Ffw7$$\"1&Q!RkFo;aF.$\"1'H-@hB1X\"F.7$F\\x7$$\"1%G.d*eo$)eF.$\"1 rm$fBA))Q\"F.7$Fbx7$$\"1vg)R#\\k*R'F.$\"15mXD4?O8F.7$Fhx7$$\"14*G,+A$o pF.$\"17u)R(*fDH\"F.7$F^y7$$\"1pP(*G#pQf(F.$\"1W#o!4\"GyD\"F.7$Fdy7$$ \"1v$3K7F.7$Fjy7$$\"1k%[`xPS.*F.$\"1*\\yu.$e:7F.7 $F`z7$$\"1,)QGyM(e)*F.$\"1BWC.uz37F.7$Ffz7$$\"1WsH\"*e.w5!#:$\"1x)yKek C@\"F.7$F\\[l7$$\"1/W\")=MXu6F_[l$\"1n(HLNewA\"F.7$Fc[l7$$\"1<*RHT#p\" G\"F_[l$\"1g/#zf%)eD\"F.7$Fi[l7$$\"1!=@7v.$)R\"F_[l$\"1/\"oV`9#*H\"F.7 $F_\\l7$$\"19+f+VzC:F_[l$\"1Wv5?2Zg8F.7$Fe\\l7$$\"1_$pAD&eh;F_[l$\"1d[ -x%)[V9F.7$F[]l7$$\"1#z7w`^*3=F_[l$\"1_:bJb[`:F.7$Fa]l7$$\"1Z5C$*R$p'> F_[l$\"1PKtjre(p\"F.7$Fg]l7$$\"1L$3\"yB@N@F_[l$\"1eJ-VTd&)=F.7$F]^l7$$ \"1(4NNbAHJ#F_[l$\"10Ys/l%48#F.7$Fc^l7$$\"1)>I@C&R)\\#F_[l$\"1%R!\\u[U _CF.7$Fi^l7$$\"1!e+6$zy)o#F_[l$\"18H$RiIg(GF.7$F__l7$$\"1x+9'[z&zGF_[l $\"1g3aFD'yV$F.7$Fe_l7$$\"1S%Qdv3R1$F_[l$\"1\"p'\\;\"Qv=%F.7$F[`l7$$\" 1i>n`dyJKF_[l$\"1(y(Reu\"=>&F.7$Fa`l7$$\"1iUv/yKpLF_[l$\"1%>\\ZWlo`'F. 7$Fg`l7$$\"1?C_`&H%eMF_[l$\"14d7D'ebK)F.7$F]al7$$\"1Ar,yQoxMF_[l$\"1M! 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y[\"e<<5F.$\"1())43T$))RCF.7$Fhifl7$$\"1[Xq=8A)4\"F.$\"1t,dyp:JAF.7$F^ jfl7$$\"1$[+e[$3)=\"F.$\"1y,&3[H?/#F.7$Fdjfl7$$\"18[*fM:wG\"F.$\"1![oq #Hqq=F.7$Fjjfl7$$\"1v^VfNw(R\"F.$\"1Q&*f/pa:: F.$\"1pcV,Q3v:F.7$Ff[gl7$$\"1$e<-9;Ul\"F.$\"1Yfvc`(zW\"F.7$F\\\\gl7$$ \"18G5\\R'H!=F.$\"1rb*)\\/,L8F.7$Fb\\gl7$$\"17QR%HPs'>F.$\"1&*HTWS4H7F .7$Fh\\gl7$$\"1ml:5+g[@F.$\"1'*[\\6sBN6F.7$F^]gl7$$\"1!H#Qoqx[BF.$\"1H Aml\"[00\"F.7$Fd]gl7$$\"1oU1j,npDF.$\"1%oVlbUAu*Fdf`l7$Fj]gl7$$\"1gt^r ZP8GF.$\"1RI]Y'[a0*Fdf`l7$F`^gl7$$\"1x/9*y)>#3$F.$\"1T\\&*p=lQ%)Fdf`l7 $Ff^gl7$$\"1&yN0$GoyLF.$\"1CvAR'3h)yFdf`l7$F\\_gl7$$\"1^T;\"GAcq$F.$\" 1Beb#zMFR(Fdf`l7$Fb_gl7$$\"1E&Rs;\"4mSF.$\"1,%y-RpS&pFdf`l-F_\\r6&Fa\\ rF*Fb\\rF*Fe\\rFi\\r-%%VIEWG6$;$!+,T>_ " 0 "" {MPLTEXT 1 0 106 "K0:=2:B:=.1:init1 := x(0)=.9, y(0)=2:ini t2 := x(0)=1,y(0)=.2:init3:= x(0)=2,y(0)=.5:init4:= x(0)=1,y(0)=1:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "modLotkaVolterra1:=\{op(mod LotkaVolterra)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2modLotkaVolter ra1G<$/-%%diffG6$-%\"xG6#%\"tGF-,(F+\"\"\"*&F+F/%\"yGF/!\"\"*&,&$F/F2F /*$,&F/F/*$-%$expG6#,$F-F2\"\"$\"\")F2$F2F2F/F+F/F2/-F(6$-F1F,F-,&F1F2 F0F/" }}}{EXCHG {PARA 281 "" 0 "" {TEXT 283 11 "Solution #1" }}{PARA 0 "" 0 "" {TEXT -1 40 "Initial conditions are x(0)=0.9, y(0)=2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "modLVsol1:=dsolve(modLotkaVo lterra1 union \{init1\},[x(t),y(t)],\ntype=numeric,output=listprocedur e):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "modxsol1:=subs(modLV sol1,x(t)): modysol1:=subs(modLVsol1,y(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([modxsol1,modysol1],0..20,0..11,title=`Prey=0 .9 Predator=2`);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7\\t7$\" \"!$\"\"*!\"\"7$$\"+4x&)*3\"!#5$\"+e=62!)F/7$$\"++&FN$\"+W *Rl%>FN7$$\"+vm=5^FN$\"+pt/0?FN7$$\"++FZ=_FN$\"+AXtY?FN7$$\"+8dhs_FN$ \"+B7'*f?FN7$$\"+D(enK&FN$\"+5BWn?FN7$$\"+QFN7$$ \"++d5[dFN$\"+bt`&*=FN7$$\"+]$fC&eFN$\"+%)\\I\"z\"FN7$$\"+iCAZfFN$\"+w 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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "B:=1:K0:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "modLotkaVolterra1:=\{op(modLotkaVolterra)\};" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%2modLotkaVolterra1G<$/-%%diffG6$-%\" xG6#%\"tGF-,(F+\"\"\"*&F+F/%\"yGF/!\"\"*&,&F/F/*$,&F/F/*$-%$expG6#,$F- F2\"\"$\"#FF2F2F/F+F/F2/-F(6$-F1F,F-,&F1F2F0F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 11 "Solution #3" }} {PARA 0 "" 0 "" {TEXT -1 36 "Initial conditions x(0)=0.9, y(0)=2." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "modLVsol1:=dsolve(modLotkaVo lterra1 union \{init1\},[x(t),y(t)],\ntype=numeric,output=listprocedur e):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "modxsol1:=subs(modLV sol1,x(t)): modysol1:=subs(modLVsol1,y(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot([modxsol1,modysol1],0..20,0..11,title=`Prey=0 .9 Predator=2`);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7_s7$\"\" !$\"\"*!\"\"7$$\"+4x&)*3\"!#5$\"+7(*H%G(F/7$$\"+F/7$$\"+e0$=C\"FS$\"++xB!*=F/7 $$\"+'R\"=\\8FS$\"+Ka6S=F/7$$\"+LA`c9FS$\"+TwR>=F/7$$\"+rI)Qc\"FS$\"+_ $oT#=F/7$$\"+3RBr;FS$\"+%)4_^=F/7$$\"+W^\"\\)=FS$\"+5Nql>F/7$$\"+zjf)4 #FS$\"+gRM`@F/7$$\"+'4;[\\#FS$\"+7Uk)p#F/7$$\"+j'y]!HFS$\"+rG%Re$F/7$$ \"+'zs$HLFS$\"+*R`T'\\F/7$$\"+0&R3a$FS$\"+rrt&)eF/7$$\"+8iI_PFS$\"+,2( G+(F/7$$\"+;d#)pRFS$\"+m/-%R)F/7$$\"+<_M(=%FS$\"+EzT25FS7$$\"+9l$*yVFS $\"+3q/$=\"FS7$$\"+4y_qXFS$\"+dsu(Q\"FS7$$\"+?5PyYFS$\"+9$ym^\"FS7$$\" +IU@'y%FS$\"+j-yb;FS7$$\"+Su0%*[FS$\"+&Gu\\!=FS7$$\"+]1!>+&FS$\"+\"ydP '>FS7$$\"+vm=5^FS$\"+M-qJ@FS7$$\"++FZ=_FS$\"+z041BFS7$$\"+D(enK&FS$\"+ oM^$[#FS7$$\"+]Z/NaFS$\"+\\\\oeEFS7$$\"++%)RRbFS$\"+\"H)4=GFS7$$\"+]?v VcFS$\"+y:TeHFS7$$\"++d5[dFS$\"+N/qmIFS7$$\"+]$fC&eFS$\"+K26FJFS7$$\"+ 14%)**eFS$\"+YL/MJFS7$$\"+iCAZfFS$\"+\\:%f7$FS7$$\"+=Sg%*fFS$\"+._S,JF S7$$\"+tb)>/'FS$\"+wdKfIFS7$$\"+&o[n8'FS$\"+iSK?HFS7$$\"+'z6:B'FS$\"+_ W^5FFS7$$\"+\\I&yG'FS$\"+f!\\ob#FS7$$\"+-V>WjFS$\"+@'3nQ#FS7$$\"+bb`+k FS$\"+%*GP0AFS7$$\"+2o(oX'FS$\"+lwa=?FS7$$\"+h!=K^'FS$\"+(pA=$=FS7$$\" +8$f&plFS$\"+#o$=];FS7$$\"+m0!fi'FS$\"+(=iwZ\"FS7$$\"+<=C#o'FS$\"+KQ;< 8FS7$$\"+\"o)pxnFS$\"+8<=x5FS7$$\"+Vb:toFS$\"+TX4*y)F/7$$\"+1ChopFS$\" +Q\\I&>(F/7$$\"+n#pS1(FS$\"+5F>OfF/7$$\"+;t9'G(FS$\"+bLPrRF/7$$\"+j`A3 vFS$\"+`c&*pGF/7$$\"+m?![q(FS$\"+](Q4I#F/7$$\"+n(y8!zFS$\"+O&Ge&>F/7$$ \"+mX0<\")FS$\"+(*GvPOI;F/7$$\"+EZ5Q&)FS$\"+e!e ;g\"F/7$$\"+)3zMu)FS$\"+)HZ+j\"F/7$$\"+!plx&*)FS$\"+-VQ8]'F/7$$\"+;*[H7\"Fj`l$\"+!eu6s(F/7$$\"+VKOW6Fj`l$\"+4#))o B*F/7$$\"+qvxl6Fj`l$\"+I5i06FS7$$\"+7ts'=\"Fj`l$\"+ni@<8FS7$$\"+`qn27F j`l$\"+'y]bc\"FS7$$\"+H?\"y@\"Fj`l$\"+\\=V*p\"FS7$$\"+0q%zA\"Fj`l$\"+M K9U=FS7$$\"+\")>3Q7Fj`l$\"+FS7$$\"+cp@[7Fj`l$\"+JhP^@FS7$$\"+>, Zf7Fj`l$\"+'3dKL#FS7$$\"+#GB2F\"Fj`l$\"+S&pw^#FS7$$\"+Xk(>G\"Fj`l$\"+( QT#)p#FS7$$\"+3'HKH\"Fj`l$\"+K\"3b'GFS7$$\"+w5M.8Fj`l$\"+,^a$*HFS7$$\" +VDX88Fj`l$\"+]*)[(3$FS7$$\"+x#3&=8Fj`l$\"+$4[q6$FS7$$\"+5ScB8Fj`l$\"+ #4XB8$FS7$$\"+x=4E8Fj`l$\"+%)[-MJFS7$$\"+W(>'G8Fj`l$\"+a*398$FS7$$\"+6 w9J8Fj`l$\"+AkFCJFS7$$\"+xanL8Fj`l$\"+EcU7JFS7$$\"+F;ZW8Fj`l$\"+*3l^+$ FS7$$\"+wxEb8Fj`l$\"+U+e.GFS7$$\"+^emg8Fj`l$\"+np+qEFS7$$\"+ER1m8Fj`l$ \"+5yH=DFS7$$\"+,?Yr8Fj`l$\"+\\X`_BFS7$$\"+v+'oP\"Fj`l$\"+GY[x@FS7$$\" +M:v\"Q\"Fj`l$\"+*Hg],#FS7$$\"+#*Hk'Q\"Fj`l$\"+zjz_=FS7$$\"+]W`\"R\"Fj `l$\"+Vt)Rp\"FS7$$\"+3fU'R\"Fj`l$\"+*4*RT:FS7$$\"+C)3iS\"Fj`l$\"+;nxi7 FS7$$\"+S<*fT\"Fj`l$\"+h1XE5FS7$$\"+_qoE9Fj`l$\"+h<>x\")F/7$$\"+jBQP9F j`l$\"+,!ebb'F/7$$\"+uw2[9Fj`l$\"+%=zvJ&F/7$$\"+&)Hxe9Fj`l$\"+f%>,Q%F/ 7$$\"+%*)**)y9Fj`l$\"+%HE5>$F/7$$\"+.o-*\\\"Fj`l$\"+*[nI[#F/7$$\"+A_1? :Fj`l$\"+9.yV?F/7$$\"+TO5T:Fj`l$\"+k_K!z\"F/7$$\"+UDnh:Fj`l$\"+@e\"yl \"F/7$$\"+U9C#e\"Fj`l$\"+o$)*\\g\"F/7$$\"+u^x.;Fj`l$\"+x#\\`h\"F/7$$\" +1*3`i\"Fj`l$\"+H^#Go\"F/7$$\"+$*zym;Fj`l$\"+)QP]'>F/7$$\"+^j?4Fj`l$\"+A.%pC\"FS7$ $\"+q`KO>Fj`l$\"+7;cu9FS7$$\"+/Uac>Fj`l$\"+XhhPFj`l$\" +8XU$*=FS7$$\"+-@Fy>Fj`l$\"+@+Te?FS7$$\"+^g8*)>Fj`l$\"+,V#4B#FS7$$\"#? 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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "restart:LotkaVolterra := [diff(x(t),t) = r*x - a*x*y , diff(y(t),t) = -m*y + b*x*y];r:=1:a:=1:m:=1:b:=1:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%.LotkaVolterraG7$/-%%diffG6$-%\"xG6#%\"tGF-,&*&%\"r G\"\"\"F+F1F1*(%\"aGF1F+F1%\"yGF1!\"\"/-F(6$-F4F,F-,&*&%\"mGF1F4F1F5*( %\"bGF1F+F1F4F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(p lots):with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "ini t1 := x(0)=.9, y(0)=2:init2 := x(0)=1,y(0)=.2:init3:= x(0)=2,y(0)=.5:i nit4:= x(0)=1,y(0)=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "P 1:=phaseportrait(LotkaVolterra,[x(t),y(t)],0..10,[[init1]],x=0..2.4,y= 0..2.6,stepsize=.1,linecolour=[blue], title=`Phaseportrait 1`):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "B:=.1:C:=1:d:=3:q:=1:K0:=2:modLotkaVolterra:=[diff(x(t),t)=r*x-a* x*y-p(t)*x,diff(y(t),t)=-m*y+b*x*y];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%1modLotkaVolterraG7$/-%%diffG6$-%\"xG6#%\"tGF-,(F+\"\"\"*&F+F/%\"y GF/!\"\"*&-%\"pGF,F/F+F/F2/-F(6$-F1F,F-,&F1F2F0F/" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 60 "f:=k->B*(1-1/(1+(C*k)^d)):K:=t->K0*exp(1-q*t ):p:=t->f(K(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "P2:=ph aseportrait(modLotkaVolterra,[x(t),y(t)],0..30,[[init1]],stepsize=.1,l inecolour=[green],title=`B=0.1, K0=2`):" }}}{EXCHG