r1:=0.99
r2:=0.32
K1:=0.75
K2:=0.97
α12:=0.51
α21:=0.55
f[x_,y_]:=r1*x[t]*((K1-x[t]-α12*y[t])/K1)
g[x_,y_]:=r2*y[t]*((K2-y[t]-α21*x[t])/K2)
x0:=0.5
y0:=0.5
tmax:=50
rjesenje:=NDSolve[{x'[t]==f[x,y],y'[t]==g[x,y],x[0]==x0,y[0]==y0},{x,y},{t,0,tmax}]
rjesenje
rj14=ParametricPlot[Evaluate[{x[t],y[t]}/.rjesenje],{t,0,tmax},PlotStyle->{{Red},{RGBColor[0.1,0.8,0.9]}}]

Show[rj14, PlotRange->{{0,0.7},{0,1}},AxesOrigin->{0,0}]