Phase Plane Analysis of Lotka-Volterra

Defining components of the vector field. Note: I have removed three of the constants by converting to dimensionless variables.

f[x_,y_]:=-x+a*x*y;
g[x_,y_]:=y-x*y;

Plotting Phase Portrait

Clear[a];
a=1/2;
plot1=ContourPlot[f[x,y]==0,{x,-.1,4},{y,-.1,4},ContourStyle->{Thick,Red},PlotPoints->40];
plot2=ContourPlot[g[x,y]==0,{x,-.1,4},{y,-.1,4},ContourStyle->{Thick,Purple},PlotPoints->40];
plot3=StreamPlot[{f[x,y],g[x,y]},{x,0,4},{y,0,4},StreamStyle->Black];
plota=Show[plot1,plot2,plot3]

Conserved Quantity

Clear[a];
a=1/2;
L[x_,y_]:=y*Exp[-a*y]*x*Exp[-x];
plot4=DensityPlot[L[x,y],{x,0,4},{y,0,4},ColorFunction->"Rainbow",PlotPoints->20];
Show[plot4,plota]