eqn=NDSolve[{x'[t]==x[t](3-2x[t]-2y[t]),y'[t]==y[t](2-2x[t]-y[t]),x[0]==1,y[0]==2},{x,y},{t,0,10}]

des=Plot[{x[t]/.eqn,y[t]/.eqn},{t,0,10}]

SetDirectory["d:/"]

Export["population2.pdf",des,"PDF"]

solveEqDif[f_,xy0_,tmax_]:=
Module[{sol,tp},
sol=NDSolve[{x'[t]==x[t](3-2x[t]-2y[t]),y'[t]==y[t](2-2x[t]-y[t]),x[0]==xy0[[1]],y[0]==xy0[[2]]},{x,y},{t,0,tmax}];
tp=sol[[1,1,2,1,1,2]];
ParametricPlot[Evaluate[{x[t],y[t]}/.sol],{t,0,tp},PlotStyle->{Magenta,Thick}]
]

Manipulate[
Show[StreamPlot[{ y(2-2x-y ),x(3-2x-2y) },{x,-1,3},{y,-1,3},ColorFunction->None],
Plot[3/2-x,{x,-3,3},PlotStyle->Red],
ParametricPlot[{0,y},{y,-3,3},PlotStyle->Red],
Plot[0,{x,-3,3},PlotStyle->Blue],
Plot[2-2x,{x,-3,3},PlotStyle->Blue],
solveEqDif[f,p,5]],{{p,{1,1}},Locator}]