Phase Portrait for a Competitive System restart: with(plots): with(DEtools): Define the differential equation using parameters: a:=2: b:=1: c:= 1: d:= 3: e:=1: f:=2: g:=2: CompDE := [diff(x(t),t) = x(t)*(a-b*x(t)-c*y(t)), diff(y(t),t) = y(t)*(d-e*x(t)-f*y(t))]; Plot the phaseportrait using Runge-Kutta method; the initial conditions are given in the first line and can be changed. IC:=[[x(0)=0.1,y(0)=0.1],[x(0)=2,y(0)=0.1],[x(0)=0.1,y(0)=1.5], [x(0)=3,y(0)=3],[x(0)= 2,y(0)=3.0],[x(0)= 3,y(0)=2.0]]; phaseportrait(CompDE, [x(t),y(t)], t=0..7, IC,linecolour=BLUE, x=0..4, y=0..4, stepsize=0.1, arrows=NONE, method=classical[rk4], title=`Competitive System`); You could try changing the parameters in the equation to get different examles. Plot x versus t a:=2: b:=1: c:= 1: d:= 3: e:=1: f:=2: g:=2: CompDE := [diff(x(t),t) = x(t)*(a-b*x(t)-c*y(t)), diff(y(t),t) = y(t)*(d-e*x(t)-f*y(t)), diff(T(t),t)=1]; IC:=[[x(0)=0.1,y(0)=0.1,T(0)=0],[x(0)=2,y(0)=0.1,T(0)=0], [x(0)=0.1,y(0)=1.5,T(0)=0],[x(0)=3,y(0)=3,T(0)=0], [x(0)= 2,y(0)=3.0,T(0)=0],[x(0)= 3,y(0)=2.0,T(0)=0]]; DEplot(CompDE, [x(t),y(t),T(t)], t=0..7, IC,linecolour=BLUE, x=0..4, y=0..4, stepsize=0.1, arrows=NONE, method=classical[rk4],scene=[T,x], title=`Competitive System`); Plot the vector field for x' = x(2-x-y), y' = y(3-x-2y) fieldplot([x*(2-x-y),y*(3-x-2*y)], x = 0..4, y=0..4); Plot the solutions with several initial conditions using Euler method: IC:=[[x(0)=0.1,y(0)=0.1,T(0)=0],[x(0)=2,y(0)=0.1,T(0)=0], [x(0)=0.1,y(0)=1.5,T(0)=0],[x(0)=3,y(0)=3,T(0)=0]]; phaseportrait(CompDE, [x(t),y(t),T(t)], t=0..7, IC,linecolour=BLUE, x=0..4, y=0..4, arrows=NONE,method=classical[foreuler],scene=[T,x], title=`Competitive System`); Notice the problem with the solution overshooting and crossing another solution. This is caused by using the Euler method and does not happen for the true solutions. 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