Two Omivores

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

f[x_,y_]:=x*(1-x)-γ*x*y;
g[x_,y_]:=η*y*(1-y)-δ*x*y;

Plotting Phase Portrait

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

Bifurcation

Clear[γ,δ,η];

Animate[plot1=ContourPlot[x*(1-x)-γ*x*y==0,{x,-.1,2},{y,-.1,2},ContourStyle->{Thick,Red},PlotPoints->40];
plot2=ContourPlot[η*y*(1-y)-δ*x*y==0,{x,-.1,2},{y,-.1,2},ContourStyle->{Thick,Purple},PlotPoints->40];
plot3=StreamPlot[{x*(1-x)-γ*x*y,η*y*(1-y)-δ*x*y},{x,0,2},{y,0,2},StreamStyle->Black];
plota=Show[plot1,plot2,plot3],{γ,1/2,2},{η,0,2},{δ,0,2},AnimationRunning->False]