Plot Point On StreamPlot

Question

How would I go about plotting a single point in the $x,y$ plane given by; $$\left(\frac{\lambda d -b\mu}{ad-bc}, \frac{a \mu -c\lambda}{ad-bc} \right)$$ Where we can take $a,b,c,d,\mu$ to be positive constants and $\lambda$ to be some parameter that is free to change. Namely, I want to have $\lambda$ in a manipulate environment and see how the point changes whilst the streamplot changes accordingly. It can be noted that in fact this is a fixed point to the system of differential equations; $$ \frac{du}{dt}=u(\lambda-au-bv)$$ $$ \frac{dv}{dt}=v(\mu-cu-dv) $$ I have currently implemented the differential equation, having the streamplot in the manipulate environment, all that is left now is to somehow plot this point on the same graph where changing the value of lambda simultaneously moves the fixed point and the streamplot surrounding it. This is what I have done so far;

        LVA[a_, b_, c_, d_, u_, l_] := {x*(l - a*x - b*y), y*(u - c*x - d*y)}
        FPStream = 
        Manipulate[
        StreamPlot[
        LVA[2, 1, 1.2, 2.1, 0.6, l], {x, -0.4, 0.4}, {y, -0.4, 0.4}, 
        PlotLabel -> Row[{"l = ", l}], GridLines -> Automatic], {l, 0, 2}]

Thank you for your help in advance!

Answer 1

In MATLAB, what I generally use, you can just use the 'hold on' command.

There is no hold on in Mathematica, but you can add a point to a plot in many ways. One is to use Epilog

Code

Clear["Global`*"];
fx[a_, b_, c_, d_, u_, L0_, x_, y_] := x*(L0 - a*x - b*y)
fy[a_, b_, c_, d_, u_, L0_, x_, y_] := y*(u - c*x - d*y);
xCoord[a_, b_, c_, d_, u_, L0_] := (L0*d - b*u)/(a*d - b*c);
yCoord[a_, b_, c_, d_, u_, L0_] := (a*u - c*L0)/(a*d - b*c);

Manipulate[
 Module[{x0, y0},

  x0 = xCoord[a, b, c, d, u, L0];
  y0 = yCoord[a, b, c, d, u, L0];

  Grid[{
    {Row[{"xCoord = ", x0, " yCoord =", y0}]},
    {StreamPlot[{fx[a, b, c, d, u, L0, x, y], 
       fy[a, b, c, d, u, L0, x, y]}, {x, -1, 1}, {y, -1, 1},
      Epilog -> {Red, PointSize[.04], Point[{x0, y0}]}
      , ImageSize -> 300, PerformanceGoal -> "Quality"
      ]
     }
    }]
  ]

 ,

 {{a, 2, "a"}, 0, 2, .1, Appearance -> "Labeled"},
 {{b, 1, "b"}, 0, 2, .1, Appearance -> "Labeled"},
 {{c, 1.2, "c"}, 0, 2, .1, Appearance -> "Labeled"},
 {{d, 2.1, "d"}, 0, 3, .1, Appearance -> "Labeled"},
 {{u, 0.6, "u"}, 0, 2, .1, Appearance -> "Labeled"},
 {{L0, 1, "L"}, 0, 2, .1, Appearance -> "Labeled"},
 TrackedSymbols :> {a, b, c, d, u, L0}
 ]