Plotting vectors in a vectorplot along edges of a plot

Question

I am plotting a vector plot that should represent trajectories through a space of values.

When the trajectories hit one of the boundaries (the values 0 or 1 on each axis), the trajectories are not supposed to be able to go any further. So at that point, the trajectories should evolve along the edge.

I am wondering how to produce this in a plot.

Currently, the vector plot doesn't 'know' that the trajectories cannot go above 1 or below 0 on each axis, so they aren't producing the trajectories I need along the edge. I'd be grateful to anyone who knows how to solve this.

x = 2;
y = 1;
k = 3;


Plota = VectorPlot[{
   
   (1/2 (x k - 
       2 y - (x - y) (-1 + 
          k) mbar + ((x^2) k - (x*y) (-1 + k) x) x) (-1 + x)),
   
   (1/2 (-1 + k) (y + x (-1 + vbar) - x vbar + 
       y (x (-1 + vbar) - y vbar)))
   
   },
  
  {vbar, 0, 1}, {mbar, 0, 1},
  
  VectorStyle -> {"Arrow", Black, Opacity[1]},
  VectorScale -> {0.01, 6, None},
  VectorPoints -> 13,
  
  StreamPoints -> 200,
  StreamScale -> {Full, All, 0.05},
  StreamStyle -> {"Line", Black, Opacity[0.5]},
  
  FrameLabel -> {"v", "m"},
LabelStyle -> Directive[ 25, Black, FontFamily -> "Calibri Light"]]

That produces this:

Answer 1

Here's a way to use VectorPlot for this purpose:

x = 2;
y = 1;
k = 3;

v[vbar_, mbar_] := {
  (1/2 (x k - 2 y - (x - y) (-1 + k) mbar + ((x^2) k - (x*y) (-1 + k) x) x) (-1 + x)),
  (1/2 (-1 + k) (y + x (-1 + vbar) - x vbar + y (x (-1 + vbar) - y vbar)))
  }
vRestricted[vbar_, mbar_] := Which[
  vbar > 0.98 || vbar < 0.02, Projection[v[vbar, mbar], {0, 1}],
  mbar > 0.98 || mbar < 0.02, Projection[v[vbar, mbar], {1, 0}],
  True, v[vbar, mbar]
  ]

VectorPlot[
  vRestricted[vbar, mbar],
  {vbar, 0, 1},
  {mbar, 0, 1},
  FrameLabel -> {"v", "m"},
  LabelStyle -> Directive[25, Black, FontFamily -> "Calibri Light"]
  ]

A caveat is that this is very sensitive to the arguments that you supply to VectorPlot. If you sample the space differently, you may miss the vertical and horizontal arrows, get multiple rows of them, or get arrows on the border that aren't perfectly aligned.

A more reliable way of doing this would be to draw the arrows along the border manually, for example using the Epilog option.

One way to take this idea further if you want to have a smooth transition from the outer behavior and the inner behavior, would be to linearly interpolate v and the projection depending on how close to the border the position is.