With[{α=1.7},s=NDSolve[{u'[t]==u[t](1-v[t]),
		  v'[t]== α v[t](u[t]-1),
		  {u[0]==2,v[0]==1}},
                      {u,v},
	            {t,20}]]

{{u->InterpolatingFunction[{{0.`,20.`}},"<>"],v->InterpolatingFunction[{{0.`,20.`}},"<>"]}}
ParametricPlot[Evaluate[{u[t],v[t]}/.s],{t,0,20},PlotRange->{{0,10},{0,3}}  ]

Manipulate[
s=NDSolve[{u'[t]==u[t](1-v[t]),
		  v'[t]== α v[t](u[t]-1),
		  u[0]==u0,v[0]==v0},
                      {u[t],v[t]},
	            {t,0,20}];
GraphicsColumn[{ParametricPlot[Evaluate[{u[t],v[t]}/.s],{t,0,20},PlotRange->{{0,10},{0,10}}],
Plot[Evaluate[{u[t],v[t]}/.s],{t,0,20},PlotRange->{{0,10},{0,3}}  ]  } ],
{{s,{}},None},
{{u0,1.5},0.1,3,0.5},{{v0,1},0.1,3,0.5},{{α,1.5},0.1,4.}]