# Exercises 4,5,6,7 in Section 6.3
# Solve for the equil points and find linearized DE for the system below
# x'=60x-4x^2-3xy, y'=42y-2y^2-3xy
# Then classify as spiral, center, node, saddle and comment on stability.
# Construct a phase portrait for each equil point within the biological domain x>0, y>0.F1:=(x,y)->60*x-4*x^2-3*x*y; F2:=(x,y)->42*y-2*y^2-3*x*y; solve({F1(x,y)=0,F2(x,y)=0},{x,y});F:=unapply(<F1(x,y),F2(x,y)>,(x,y)):F(x,y);J:=unapply(<diff(F1(x,y),x),diff(F2(x,y),x)|diff(F1(x,y),y),diff(F2(x,y),y)>,(x,y)):J(x,y);with(LinearAlgebra):X:=[0,0,15,6]:Y:=[0,21,0,12]:for i from 1 to 4 do 'J'(X[i],Y[i]) =J(X[i],Y[i]); od;ss:='ss':for i from 1 to 4 do ss[i]:=Eigenvalues(J(X[i],Y[i])):od:eval(ss);
for i from 1 to 4 do printf("Eigenvalues J(%a,%a) = %g, %g\134n",X[i],Y[i],ss[i][1],ss[i][2]) od:for i from 1 to 4 do <dx,dy> = J(X[i],Y[i]).<x,y> od;# Classifications done from Caley-Hamilton-Zeibur examples for x(t), y(t)# unstable node, stable node, stable node, saddle# Phase portraits are done from maple as follows:
# Tools ==> Tasks (Browse) ==> Differential Equations ==> ODEs ==> Phase Portrait
# Then click on INSERT DEFAULT CONTENT, followed by editing the data boxes.# Alternative launch into the program: ?Task,phaseportrait
# Then click on INSERT DEFAULT CONTENT at the top of the help file.TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5Mzg1NjQ4OTU4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjeik2JCUieEclInlHW1tbW1tfW1tbW3MhIiVcW1tbW11bW1tbcyEiJFtbW1tbXVtbW1tvIiNneilGJ1xbW1tbXVtbW1tzRitdW1tbW1tbW1tbcyEiI1xbW1tbW1tbW1tvIiNVRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5Mzg1NjQ5NDM4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsKCUieEchIiklInlHISIkIiNnIiIiLCRGKUYqLCRGJ0YqLChGJ0YqRikhIiUiI1VGLEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5Mzg1NjQ5NTU4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiI2ciIiFGJyIjVUYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5Mzg1NjQ5Njc4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMhIiQhI2oiIiEhI1VGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5Mzg1NjQ5Nzk4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMhI2ciIiEhI1ghIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5Mzg1NjQzMTgyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMhI0MhI08hIz1GJkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzIwMTk4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiNnIiNVRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzE1OTgyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjISIkISNVRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzEyMTI2WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjISIkISNnRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzA3MDcwWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCYhI0MiIiIqJCIiIyNGKEYqIiM9LCZGJ0YoRikhIz1GJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzA3MTkwWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSNkeEclI2R5R0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzA3NTUwWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCQlInhHIiNnLCQlInlHIiNVRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzA3NjcwWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSNkeEclI2R5R0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzAxNDE0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCQlInhHISIkLCZGJyEjaiUieUchI1VGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzAxNTM0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSNkeEclI2R5R0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzAyMDE0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCYlInhHISNnJSJ5RyEjWCwkRikhIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzAyMTM0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJSNkeEclI2R5R0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2ODgzODY5MzgyNzAzMjE0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCYlInhHISNDJSJ5RyEjPSwmRichI09GKUYoRiU=