sol[a_,b_]:=NDSolve[{x'[t]==x[t]-x[t]^2-x[t]y[t],
y'[t]==1/2y[t]-1/4y[t]^2-3/4x[t]y[t],
x[0]==a,y[0]==b},{x,y},{t,20}]

s1=Table[ParametricPlot[Evaluate[{x[t],y[t]}/.sol[2.0,b]],{t,0,20},
AspectRatio-> 1,PlotRange->{{0,2},{0,2}}],
{b,0.0,2.0,0.2}];

s2=Table[ParametricPlot[Evaluate[{x[t],y[t]}/.sol[a,2.0]],{t,0,20},
AspectRatio-> 1,PlotRange->{{0,2},{0,2}}],
{a,0.,2.0,0.2}];

s3=Table[ParametricPlot[Evaluate[{x[t],y[t]}/.sol[c,0.2-c]],{t,0,20},
AspectRatio-> 1,PlotRange->{{0,2},{0,2}}],
{c,0.0,0.20,0.02}];

Show[s1,s2,s3]

Animate[ParametricPlot[Evaluate[{x[t],y[t]}/.sol[2.0,b]],{t,0,20},
PlotRange->{{0,2},{0,2}}],
{b,0.0,2.0,0.1},AnimationRunning->False]

Animate[ParametricPlot[Evaluate[{x[t],y[t]}/.sol[c,0.2-c
]],{t,0,20},
AspectRatio-> 1,PlotRange->{{0,2},{0,2}}],
{c,0.0,0.20,0.01}]