f[u_,v_]:=u[t](1-v[t])
g[u_,v_,α_]:=α v[t](u[t]-1)

F[u_,v_]:=u(1-v)
G[u_,v_,α_]:=α v(u-1)
xmax:=4
ymax:=4

Manipulate[
s=NDSolve[{u'[t]==f[u,v],
		  v'[t]==  g[u,v,α],
		  u[0]==u0,v[0]==v0},
                      {u[t],v[t]},
	            {t,0,20}];
pion=ContourPlot[F[x,y]==0,{x,-0.1,xmax},{y,-0.1,ymax},ContourStyle->{Green}];
poziom=ContourPlot[G[x,y,α]==0,{x,-0.1,xmax},{y,-0.1,ymax},ContourStyle->{Blue}];
vf=VectorPlot[{F[x,y],G[x,y,α]}, {x,0,xmax},{y,0,ymax},VectorScale->{0.1,0.5},VectorPoints->15,VectorStyle->{GrayLevel[0.8]}];
trajektorie = ParametricPlot[Evaluate[{u[t],v[t]}/.s],{t,0,20},PlotRange->{{0,xmax},{0,ymax}}];
pStac=NSolve[{F[x,y]==0,G[x,y,α]==0},{x,y}];
 
GraphicsColumn[
{
Show[{vf,pion,poziom,trajektorie,Graphics[{Red,PointSize[Large],Tooltip[Point[{x,y}],Round[{x,y},.2]]/.pStac}] }], 
Plot[Evaluate[{u[t],v[t]}/.s],{t,0,20},PlotRange->{{0,20},{0,4}}  ]  
}
],
{{s,{}},None},
{{u0,1.5},0.1,3,0.5},{{v0,1},0.1,3,0.5},{{α,1.5},0.1,4.}]