The Lotka-Volterra model If you click on !!! (the menu bar at the top) maple will execute all commands. You can also enter each command as you go along reading this document. 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-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6d`m-I#miGF$6#Q!F'-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.0emF'/%&depthGF4/%*linebreakGQ(newlineF'-F06&F2F5F8/F;Q%autoF'-F,6%Q(restartF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q":F'/FHQ'normalF'/%&fenceGQ&falseF'/%*separatorGFR/%)stretchyGFR/%*symmetricGFR/%(largeopGFR/%.movablelimitsGFR/%'accentGFR/%'lspaceGQ,0.2777778emF'/%'rspaceGF[oF/F=-F,6%Q%withF'FDFG-I(mfencedGF$6$-F#6#-F,6%Q.LinearAlgebraF'FDFGFNFJF/F=-FK6-Q"~F'FNFPFSFUFWFYFenFgn/FjnQ&0.0emF'/F]oF]pFioF^o-Fbo6$-F#6#-F,6%Q&plotsF'FDFGFNFJF/F=FioFioFioFioF^o-Fbo6$-F#6#-F,6%Q/VectorCalculusF'FDFGFNFJF/F=F/F=-F,6%Q#LVF'FDFGFio-FK6-Q*&coloneq;F'FNFPFSFUFWFYFenFgnFinF\oFio-FK6/Q%procF'/%%boldGFF/FHQ%boldF'/%+fontweightGFiqFPFSFUFWFYFenFgnF\pF^pFio-Fbo6$-F#6+-F,6%Q"KF'FDFG-FK6-Q",F'FNFP/FTFFFUFWFYFenFgnF\p/F]oQ,0.3333333emF'-F,6%Q"rF'FDFGFcr-F,6%Q"sF'FDFGFcr-F,6%Q"uF'FDFGFcr-F,6%Q"vF'FDFGFNFio-FK6/Q'optionF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pFio-F,6%Q)operatorF'FDFG-FK6-Q";F'FNFPFfrFUFWFYFenFgnF\pF\oFioF/F=-FK6/Q,descriptionF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pFio-I#msGF$6#Q8Lotka-Volterra~functionF'F[tFioF/F=-FK6/Q'returnF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pF/F=-F,6%Q"PF'FDFG-FK6-Q(&srarr;F'FNFPFSFUFWFYFenFgnFinF\o-Fbo6&-F#6<-Fbo6$-F#6'-I#mnGF$6$Q"1F'FN-FK6-Q"+F'FNFPFSFUFWFYFenFgn/FjnQ,0.2222222emF'/F]oF^vFir-FK6-Q"*F'FNFPFSFUFWFYFenFgn/FjnQ,0.1666667emF'/F]oFdv-Fbo6$-F#6(Ffu-FK6-Q*&uminus0;F'FNFPFSFUFWFYFenFgnF]vF_vFht-Fbo6&-F#6#FfuFN/%%openGQ"[F'/%&closeGQ"]F'-FK6-Q"/F'FNFPFS/FVFFFWFYFenFgnFcvFevF`rFNFNF`vFhtF]wFjvF\sF`vFhtF]wF`vFht-Fbo6&-F#6#-Fgu6$Q"2F'FNFNFawFdwFcrFio-Fbo6$-F#6%FfuFjvF_sFNF`vFhtF[xFjuFbsF`vFhtF]wF`vFhtF[xFN/FbwQ'&lang;F'/FewQ'&rang;F'F[tF/F=-FK6/Q$endF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pFioFcqFJF/F=F/F=-FK6-Q"#F'FNFPFSFUFWFYFenFgnF\pF^p-F,6%Q&~F~:=F'FDFGF/F=F]y-F,6%Q;~F:=LV(1,1.3,0.5,0.7,1.6):F'FDFGF/F=F]y-F,6%Q,~~F(<2,2>);F'FDFGF/F=F/F=F/F=-F,6%Q#ICF'FDFGF`qFcq-Fbo6$-F#6--F,6%Q#K1F'FDFGFcrFirFcrF\sFcr-F,6%Q#K2F'FDFGFcrF_sFcrFbsFNF[tF/F=F^tFio-Fbt6#QDInter-specific~competition~functionF'F[tFioF/F=FetF/F=FhtF[u-Fbo6&-F#66FhtF]wFjuFirF`vFhtF]wF`v-Fbo6$-F#6'FfuFjv-Fbo6$-F#6)FhtF]wFjuF\sF`vFhtF[xFNFgwF`zFNFcrFioFhtF[xFjuF_sF`vFhtF[xF`v-Fbo6$-F#6'FfuFjv-Fbo6$-F#6)FbsF`vFhtF]wFjuFhtF[xFNFgwFczFNFNFfxFhxF[tF/F=FjxFioFcqFJF/F=F/F=-F,6%Q(iterateF'FDFGFioF`qFioFcqFio-Fbo6$-F#6)-F,6%Q"FF'FDFGFcrFio-F,6%Q#P0F'FDFGFcrFio-F,6%Q"tF'FDFGFNF+F/F=FioFioF^tFio-Fbt6#Q@compute~the~tth~iterate~F^t(P0)F'F[tFioF/F=FesFio-F,6%Q)rememberF'FDFGF[tFioF/F=-FK6/Q#ifF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pFioFj\lFio-FK6-Q&&leq;F'FNFPFSFUFWFYFenFgnFinF\oFio-Fgu6$Q"0F'FNFio-FK6/Q%thenF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pFioFetFioFg\lFio-FK6/Q%elseF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pFioFetFioFd\l-Fbo6$-F#6$F]\l-Fbo6$-F#6+Fd\lFcrFioFg\lFcrFioFj\lFjvFfuFNFNFioFjxFioFc]lF[tF/F=FjxFioFcqFJF/F=F/F=F]y-F,6%Q4iterate(F,<2,2>,2);F'FDFGF/F=F/F=-F,6%Q&orbitF'FDFGFioF`qFioFcqFio-Fbo6$-F#6)Fd\lFcrFioFg\lFcrFio-F,6%Q"TF'FDFGFNFio-FK6/Q&localF'FfqFhqFjqFPFSFUFWFYFenFgnF\pF^pFio-F,6%Q"iF'FDFGF[tFioF/F=F^tFio-Fbt6#QPthe~orbit~of~the~point~P0~under~T~iterates~of~FF'F[tF/F=-F,6%Q$seqF'FDFG-Fbo6$-F#6/F]\l-Fbo6$-F#6)Fd\lFcrFioFg\lFcrFioFj_lFNFcrFioFj_lFio-FK6-Q"=F'FNFPFSFUFWFYFenFgnFinF\oFioFi]lFio-FK6-Q#..F'FNFPFSFUFWFYFenFgnF]vF^pFioFd_lFNFioF/F=FioFioFjxFioFcqFJFioF/F=F/F=F]y-F,6%Q8~an~F-orbit~of~length~5F'FDFGF/F=F]y-F,6%Q3~orbit(F,<2,2>,5);F'FDFGF/F=F/F=F/F=F]y-F,6%QS~the~timeplot,~multigraph,~of~a~2-population~modelF'FDFGF/F=-F,6%Q)timeplotF'FDFGF`qFcq-Fbo6$-F#6'Fd\lFcrFg\lFcrFd_lFNFioFg_lFio-F,6%Q$orbF'FDFGF[tFioF/F=F^tFio-Fbt6#QDTimeplot~of~the~two~components~in~FF'F[tFioF/F=FablF`qF]_lF]blF[tFioF/F=-F,6%Q%plotF'FDFG-Fbo6$-F#6A-Fbo6&-F#6+-Fbo6&-F#6$F``l-Fbo6$-F#6)-Fbo6&-F#6'-F,6%Q"nF'FDFGFcrFabl-Fbo6&-F#6#F^dlFNFawFdwF]wFNFawFdwFcrF^dlF[alFfuF^alFd_lFNFNFawFdwFcr-Fbo6&-F#6$F``l-Fbo6$-F#6)-Fbo6&-F#6'F^dlFcrFablFadlF[xFNFawFdwFcrF^dlF[alFfuF^alFd_lFNFNFawFdwFcrF/F=FbclFcrFedlFNFawFdwFcrF^dlF[alFfuF^alFd_lFcrF/F=-F,6%Q&styleF'FDFGF[al-Fbo6&-F#6)-F,6%Q&pointF'FDFGFcrFhelFcr-F,6%Q%lineF'FDFGFcrF[flFNFawFdwFcrF/F=-F,6%Q&colorF'FDFGF[al-Fbo6&-F#6)-F,6%Q&blackF'FDFGFcrFeflFcr-F,6%Q%blueF'FDFGFcr-F,6%Q$redF'FDFGFNFawFdwFcr-F,6%Q*thicknessF'FDFGF[alF_xFcr-F,6%Q+symbolsizeF'FDFGF[al-Fgu6$Q#14F'FNFcr-F,6%Q'labelsF'FDFGF[al-Fbo6&-F#6'F+-Fbt6#Q%timeF'Fcr-Fbt6#Q+PopulationF'F+FNFawFdwFNF+F/F=FioFioFjxFioFcqFJF/F=F/F=F]y-F,6%Q9~timeplot(F,<1,0.1>,40);F'FDFGF/F=F]y-F,6%QR~timeplot(LV(20,~.5,~0.05,~.2,~0.05),<10,2>,100);F'FDFGF/F=F]y-F,6%Qfn~timeplot(LV(35,~0.05,~0.025,~0.025,~0.032),~<10,~2>,~1000)F'FDFGF/F=F/F=-F,6%Q)phasediaF'FDFGF`qFcqF]blFioFg_lFioFablF[tF/F=FioFioFioF^tFio-Fbt6#Q=The~orbit~in~the~phase~spaceF'F[tF/F=FablF`qF]_lF]blF[tF/F=-F,6%Q&arrowF'FDFG-Fbo6$-F#6>-Fbo6&-F#6$F``l-Fbo6$-F#6+-Fbo6&-F#6'Fabl-Fbo6&-F#6#Fj_lFNFawFdwFcrFabl-Fbo6&-F#6%Fj_lFjuFfuFNFawFdwFNFawFdwFcrFj_lF[alFfuF^alFd_lFjvFfuFNFN/FbwQ"|frF'/FewQ"|hrF'Fcr-F,6%Q+differenceF'FDFGF[al-F,6%FFFDFGFcrFio-F,6%Q&shapeF'FDFGF[alFcilFcrFioF/F=FioFioFioFioFioFioF^flF[alF[glFcrFioFgglF[al-Fbo6&-F#6%-F,6%Q$P_1F'FDFGFcr-F,6%Q$P_2F'FDFGFNFawFdwFNF[tFioF/F=FjxFioFcqFJF/F=F/F=F]y-F,6%Q+~one~orbitF'FDFGF/F=F]y-F,6%Q7~phasedia(F,<1,1>,40);F'FDFGF/F=F]y-F,6%Q9~phasedia(F,<1,0.1>,40);F'FDFGF/F=F/F=F]y-F,6%Q0~several~orbitsF'FDFGF/F=F]y-F,6%Qgo~display(seq(phasedia(F,P0,40),P0=[<1,1>,<1,0.1>,~<0.2,~0.3>,~<0.2,~1.6>]));F'FDFGF/F=F/F=F]y-F,6%Q8~animation~of~one~orbitF'FDFGF/F=F]y-F,6%Qhn~display(seq(phasedia(F,<1,0.1>,T),T=1..75),insequence=true);F'FDFGF/F=F/F=-F,6%Q'growthF'FDFGF`qFcq-Fbo6$-F#6'Fd\lFcr-F,6%Q%PmaxF'FDFGFcr-F,6%Q%QmaxF'FDFGFNF+F/F=FioFioF^tFio-Fbt6#QhnThe~~growth~function~|+~~~~~~~~~~~~~~~for~a~2-population~modelF'F[tF/F=Fcil-Fbo6$-F#6I-Fbo6&-F#6$F``l-Fbo6$-F#61F``l-Fbo6$-F#62-Fbo6&-F#6'-Fbo6&-F#6)Fj_lF`v-Fgu6$Q$0.1F'FNFcr-F,6%Q"jF'FDFGF`vFa`mFNFfxFhxFcrFioFd\l-Fbo6$-F#6#F]`mFNFNFawFdwFcrFioF/F=Fj_lFioF[alFioFi]lFioF^alFio-Fgu6$Q#10F'FNF`vF`^mFNFcrFioFd`mFioF[alFioFi]lFioF^alFioF[amF`vFc^mFNFNF^[mF`[mFcrFioF/F=FioFioFioFioFioFioFio-F,6%Q(scalingF'FDFGFioF[alFio-F,6%Q.UNCONSTRAINEDF'FDFGFcrFioFb[mF[alFe[mFcrFg[mF[alFcilFcrFioF/F=FioFioFioFioFioFioF^flF[alFhflFNF[tF/F=FjxFioFcqFJF/F=F/F=F/F=F/F=-F,6%Q)dirfieldF'FDFGF`qFcqF\^mF+F/F=FioFioF^tFio-Fbt6#QinThe~directional~field~|+~~~~~~~~~~~~~~~for~a~2-population~modelF'F[tF/F=Fcil-Fbo6$-F#6OF]_mFcrFioF/F=FioFioFioFioFioFio-F,6%Q'lengthF'FDFGFioF[alFio-Fgu6$Q$.05F'FNFcrFioF^amFioF[alFioFaamFcrFioFb[mF[alFe[mFcrFg[mF[alFcilFcrFioF/F=FioFioFioFioFioFioF^flF[alFhflFNF[tF/F=FjxFioFcqFJF/F=F/F=F/F=F/F=F]y-F,6%Q9~the~growth~of~one~modelF'FDFGF/F=F]y-F,6%Q0~growth(F,1,2);F'FDFGF/F=F/F=F]y-F,6%QD~the~growth~together~with~one~orbitF'FDFGF/F=F]y-F,6%QQ~display([growth(F,1,2),~phasedia(F,<1,1>,40)]);F'FDFGF/F=F/F=F/F=F]y-F,6%QI~the~growth~together~with~several~orbitsF'FDFGF/F=F]y-F,6%Q:~display([growth(F,1,2),~F'FDFGF/F=F]y-F,6%Qin~~~~~seq(phasedia(F,P0,40),~P0=[<1,1>,<1,0.1>,~<0.2,~1.6>])]);F'FDFGF/F=F/F=F]y-F,6%QR~how~to~get~an~animation~and~the~growth~function?F'FDFGF/F=F]y-F,6%Qgodisplay([growth(F,1,2),~seq(phasedia(F,<1,1>,t),t=1..20),~insequence=true]);F'FDFGF/F=F]y-F,6%Qfodisplay(seq([growth(F,1,2),~phasedia(F,<1,1>,t)],t=1..20),insequence=true);F'FDFGF/F=F/F=F]y-F,6%Q>~equi2(LV(20,1,0.5,0.5,0.3));F'FDFGF/F=F/F=F/F=-F,6%Q&equi2F'FDFGF`qFcq-Fbo6$-F#6#Fd\lFNFioFg_lFio-F,6%Q(equiptsF'FDFGFcrFio-F,6%Q"JF'FDFGFcrFio-F,6%Q#dmF'FDFGFcrFio-F,6%Q#EVF'FDFGF[tF/F=F^tFio-Fbt6#Qfnfind~equilibria~of~2-population~model~F~and~their~stabilityF'F[tF/F=F\emFioF`qFioF/F=FioFio-F,6%Q&solveF'FDFG-Fbo6$-F#6&-Fbo6&-F#64Fd\l-Fbo6$-F#6#FhtFNF]wFioF[alFioFhtF]wFcrFioFd\lFffmF[xFioF[alFioFhtF[xFNF^[mF`[mFcrFio-Fbo6&-F#6(FhtF]wFcrFioFhtF[xFNFawFdwFNF[tF/F=F_emFioF`qFio-F,6%Q)JacobianF'FDFG-Fbo6$-F#6'Fd\lFffmFcrFioFjfmFNF[tF/F=FbemF`qF``l-Fbo6$-F#6+-F,6%Q%subsF'FDFG-Fbo6$-F#6&-F,6%Q"EF'FDFGFcrFioF_emFNFcrFioF`hmFioF[alFioF\emFNF[tF/F=FeemF`qF``l-Fbo6$-F#6(-F,6%Q,EigenvaluesF'FDFG-Fbo6$-F#6#-F,6%Q"dF'FDFGFNFcrF^imF[alFbemFNF[tF/F=F]y-F,6%Qjnprint(The~population~model~with~projection~function),~print(F);F'FDFGF/F=-F,6%Q&printF'FDFG-Fbo6$-F#6'-F,6%Q$TheF'FDFGFio-F,6%Q+equilibriaF'FDFGFio-F,6%Q$areF'FDFGFNFcrFioF/F=FioFioFdim-Fbo6$-F#6$FioF\emFNF[tF/F=Fdim-Fbo6$-F#6'F[jmFio-F,6%Q,eigenvaluesF'FDFGFioFajmFNFcrFioFdim-Fbo6$-F#6#FeemFNF[tF/F=Fdim-Fbo6$-F#6/F[jmFio-F,6%Q)absoluteF'FDFGFio-F,6%Q'valuesF'FDFGFio-F,6%Q#ofF'FDFGFio-F,6%Q$theF'FDFGFioF\[nFioFajmFNFcrFioF/F=FioFioFdim-Fbo6$-F#6%FioF``l-Fbo6$-F#6(-F,6%Q$absF'/FEFRFN-Fbo6$-F#6#-F,6%Q)&lambda;F'F^]nFNFNFcrFc]nF[alFeemFNFNF[tFioF/F=FjxFioFcqFJF/F=F/F=F+ The Lotka-Volterra model is a 2-population model for prey-predator interaction. In this model, the prey population (P_1) in itself follows logistic growth with parameters (K,r)=(1, 1.3) and the predator population (P_2) in itself follows exponential growth (actually, decay) with parameter u=1-0.7 = 0.3. In addition, there is an interaction term proportional to the number of prey times the number of predators. This interaction has a negative influence on the prey population and a positive influence on the predator population. The projection function is 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 The P_1-isoclines QyQtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLyYtLUkjTFZHRig2J0kiS0dGKEkickdGKEkic0dGKEkidUdGKEkidkdGKDYjSSJQR0YoNiMiIiImRjZGNzckRjkmRjY2IyIiI0Y4 are the vertical line P_1=0 and the line through (K,0) and r/s. The P_2 isoclines QyQtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLyYtLUkjTFZHRig2J0kiS0dGKEkickdGKEkic0dGKEkidUdGKEkidkdGKDYjSSJQR0YoNiMiIiMmRjZGNzckJkY2NiMiIiJGOUY9 are the horizontal line P_2=0 and the vertical line P_1=u/v. There are two cases to consider Case 1: K>u/v There is an equilibrium in the first quadrant QyQtSS1pbXBsaWNpdHBsb3RHNiI2JzcmLyZJIlBHRiU2IyIiIiIiIS8tSSIrRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGMkkoX3N5c2xpYkdGJTYkRiktSSIqR0YxNiQtRjo2JCQiIiYhIiItSSIvR0Y2NiMkIiM4RkAmRio2IyIiI0YsL0YpLUY6NiQkIiIoRkAtRkI2IyQiIztGQC9GRkYtL0YpO0YtJCIjNkZAL0ZGO0YtJCIjRkZAL0kmY29sb3JHRiU3JkklYmx1ZUdGJUZobkkkcmVkR0YlRmluL0klYXhlc0dGJUklbm9uZUdGJUYs Case 2: K<u/v There is no equilibrium in the first quadrant QyQtSS1pbXBsaWNpdHBsb3RHNiI2JzcmLyZJIlBHRiU2IyIiIiIiIS8tSSIrRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGMkkoX3N5c2xpYkdGJTYkLUkiKkdGMTYkLUkiL0dGNjYjJCIjNyEiIkYpLUY6NiQtRjo2JCQiIiZGQS1GPTYjJCIjOEZBJkYqNiMiIiNGLC9GKS1GOjYkJCIiKEZBLUY9NiNGRi9GTEYtL0YpO0YtJCIjOkZBL0ZMOy1JIi1HRjFGVSIiJC9JJmNvbG9yR0YlNyZJJWJsdWVHRiVGXW9JJHJlZEdGJUZeby9JJWF4ZXNHRiVJJW5vbmVHRiVGLA== The equilibria are the crossings between a P_1- and a P_2-isocline 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 Case 1 Here is an example of a typical projection function with an equilibrium in the first quadrant. The parameters are (K,r,s,u,v)=(1, 1.3, 0.5, 0.7, 1.6). QyQ+SSJGRzYiLUkjTFZHRiU2JyIiIiQiIzghIiIkIiImRiwkIiIoRiwkIiM7RixGKQ== The equilibria for this particular model are QyQtSSZlcXVpMkc2IjYjSSJGR0YlIiIi The equilibrium in the first quadrant is stable and the equilibria on the axes are saddle points. Here is the time plot for prey (blue) and predator (red) over 75 generations with initial population vector <1, 0.1> 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 Note that the population vector exhibits a damped oscillation towards an equilibrium. We may also visualize the model by drawing a vector from each point (P_1,P_2) in the plane to F(P_1,P_2). This is the growth field. The population orbit starting at a particular point is found by jumping from arrowhead to next arrowhead. 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 In the next picture the vectors are normalized that they all have the same length. This is called the directional field for F. QyQtSShkaXNwbGF5RzYiNiQtSSlkaXJmaWVsZEdGJTYlSSJGR0YlIiIiJCIjQSEiIi1JLWltcGxpY2l0cGxvdEdGJTYnNyYvJkkiUEdGJTYjRisiIiEvLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjxJKF9zeXNsaWJHRiU2JEY0LUkiKkdGOzYkLUZENiQkIiImRi4tSSIvR0ZANiMkIiM4Ri4mRjU2IyIiI0YrL0Y0LUZENiQkIiIoRi4tRks2IyQiIztGLi9GT0Y3L0Y0O0Y3RisvRk87RjdGLC9JJmNvbG9yR0YlNyZJJWJsdWVHRiVGXW9JJHJlZEdGJUZeby9JJWF4ZXNHRiVJJW5vbmVHRiVGKw== The population orbits through various initial population phases follow the directional field towards the stable equilibrium. (This is called a phase plot of the model. 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 Here we trace the population orbit through <1,0.1> 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 Here is another example of a time plot and a phase plot of a Lotka-Volterra model where the difference in phase between prey and predators is more apparent 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJHNlcUYnRi9GMi1GNjYkLUYjNigtRiw2JVEpcGhhc2VkaWFGJ0YvRjItRjY2JC1GIzYnLUYsNiVRIkZGJ0YvRjItRjs2LVEiLEYnRj5GQC9GREYxRkVGR0ZJRktGTUZPL0ZTUSwwLjMzMzMzMzNlbUYnLUYsNiVRI1AwRidGL0YyRl9vLUkjbW5HRiQ2JFEkMzAwRidGPkY+Rl9vRmVvLUY7Ni1RIj1GJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjc3Nzc3OGVtRicvRlNGYHAtRjY2Ji1GIzYkLUY2NiYtRiM2JS1GaW82JFEjMTBGJ0Y+Rl9vLUZpbzYkUSIyRidGPkY+LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJy1GLDYjUSFGJ0Y+L0ZhcVEiW0YnL0ZkcVEiXUYnRj5GPi1GOzYtUSI7RidGPkZARmJvRkVGR0ZJRktGTUZPRmFw The equilibria of this model are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmZXF1aTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJGRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYn There are two saddle points on the axes and one stable equilibrium in the first quadrant Here is a Lotka-Volterra model that seems to follow an eternal cycle. I do not know if this is an artifact of the computer or there really can be cycles in these models. 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 The equilibria of this model are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmZXF1aTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJGRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYn There are two saddle points on the axes and one unstable equilibrium in the first quadrant. Let P^* be the equilibrium in the first quadrant. These example indicate that - if P^* is stable then the population orbits will spiral towards P^* - if P^* is unstable then the population orbits are closed curves circling P^* Case 2 There is no equilibrium in the first quadrant. From a biological perspective, this case is not relevant. From a mathematical perspective, there is no difference between case 1 and case 2. QyQ+SSJGRzYiLUkjTFZHRiU2JyQiIzchIiIkIiM4RiskIiImRiskIiIoRitGLiIiIg== QyQtSSZlcXVpMkc2IjYjSSJGR0YlIiIi QyQtSSdncm93dGhHNiI2JUkiRkdGJSQiIzohIiJGKCIiIg== QyQtSShkaXNwbGF5RzYiNiM3JC1JKWRpcmZpZWxkR0YlNiVJIkZHRiUkIiM6ISIiJCIjNkYuLUktaW1wbGljaXRwbG90R0YlNic3Ji8mSSJQR0YlNiMiIiIiIiEvLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRj9JKF9zeXNsaWJHRiU2JC1JIipHRj42JC1JIi9HRkM2IyQiIzdGLkY2LUZHNiQtRkc2JCQiIiZGLi1GSjYjJCIjOEYuJkY3NiMiIiNGOS9GNi1GRzYkJCIiKEYuLUZKNiNGUi9GWEY6L0Y2O0Y6RiwvRlg7RjpGVi9JJmNvbG9yR0YlNyZJJWJsdWVHRiVGZG9JJHJlZEdGJUZlby9JJWF4ZXNHRiVJJW5vbmVHRiVGOQ== QyQtSShkaXNwbGF5RzYiNiQtSSlkaXJmaWVsZEdGJTYlSSJGR0YlJCIjOiEiIiQiIzZGLS1JJHNlcUclKnByb3RlY3RlZEc2JC1JKXBoYXNlZGlhR0YlNiVGKkkjUDBHRiUiIz8vRjc3JC1JJDwsPkdGJTYkJCIiJ0YtJCIiJUYtLUY8NiQkIiIiRi0kIiM1Ri1GRQ== JSFH