How to show a solution curve in my phase portrait?

Question

I have this dynamical system

StreamPlot[{2 x - 8/5 x^2 - x*y, 5/2 y - y^2 - 2 x*y}, {x, 0, 2}, {y, -.5, .5}]

There is a stable fixed point at (1.25, 0) (I checked through the linearized matrix). It seems to be a very weak stable fixed point. However, I have a hard time seeing any steamline going directly to that fixed point. Can anyone please help me create one solution curve (make it in red please) that goes from any positive initial conditions x and y (let's just pick x = 10, y = .05) to the fixed point (1.25, 0)? Thank you so much!

Answer 1

Something like this?

 f[{x_, y_}] := {2 x - 8/5 x^2 - x y, 5/2 y - y^2 - 2 x y}
 field = StreamPlot[f[{x, y}], {x, 0, 2}, {y, -.5, .5}];
 {xsoln, ysoln} = NDSolveValue[{{x'[t], y'[t]} == f[{x[t], y[t]}], 
      x[0] == 10, y[0] == 0.5}, {x, y}, {t, 0, 10}];
 curve = ParametricPlot[{xsoln[t], ysoln[t]}, {t, 0, 10}, 
      PlotStyle -> {Red, Thickness[Medium]}, PlotRange -> All];
 pt = {1.25, 0};
 Show[{field, curve, Graphics[{Black, PointSize[Large], Point[pt]}]}]

For the particular solution curve, I chose an initial condition a bit further away from the x-axis so it would display better.