Extra restrictions in best response functions using Manipulate[ ]?

Question

I have two best response functions that I have defined like this:

BRc[p_, s_, l_] := (3 p + 2 s + 4 l p s + 4 l s^2)/(4 (1 + 2 l s));
BRp[c_, s_, l_] := (-(c/s) - c/(s (2 + 4 l s)))/(
  1/(2 (1 + 2 l s)^2) - 2/(1 + 2 l s) - 2/(s (1 + 2 l s)));

I want to define certain ranges, values and add a title for the plot, like this:

Manipulate[
 ParametricPlot[{{p, BRc[p, s, l]}, {BRp[c, s, l], c}}, {c, 0, 
   50}, {p, 0, 50}, PlotStyle -> {{Blue, Thick}, {Red, Thick}}, 
  PlotRange -> {{0, 50}, {0, 50}}, AxesLabel -> {c, p}, 
  AspectRatio -> 1, 
  PlotLabel -> Style["Best Response Curves", Blue, 20]], 
 Style["Parameters", Bold, Medium], {s, 0, 10}, {l, 0, 10}]

But I need to define the following restrictions inside Manipulate[ ]:

p > 0 && ((0 < l < 1/
      2 && ((p < c < p/(2 l) && 
         0 < s < (-2 c + p)/(4 c l - 2 p)) || (c >= p/(2 l) && 
         s > 0))) || (1/2 <= l <= 1 && c > p && s > 0))

To make sure that the two functions always intersect.

I don't know where to insert the restrictions. Help needed here. Also I would like to name the axis

When I run the code without these restrictions, I get the following image/error:

Answer 1

I am not confident that the restrictions

p > 0 && c > 
  p && ((0 < l < p/(2 c) && 
     0 < s < (-2 c + p)/(4 c l - 2 p)) || (p/(2 c) <= l <= 1 && 
     s > 0))

will ensure that the two functions will intersect.

However to answer the question about how to implement these restrictions in Manipulate we will redefine the two functions so that each have all four of the variables c, p, s and l.

Further we will apply the restriction to the definition of the functions:

BRc[p_, s_, l_, c_] := (3 p + 2 s + 4 l p s + 4 l s^2)/(4 (1 + 2 l s)) /; 
  p > 0 && c > 
    p && ((0 < l < p/(2 c) && 
       0 < s < (-2 c + p)/(4 c l - 2 p)) || (p/(2 c) <= l <= 1 && 
       s > 0))

BRp[c_, s_, l_, p_] := (-(c/s) - c/(s (2 + 4 l s)))/(1/(2 (1 + 2 l s)^2) - 
     2/(1 + 2 l s) - 2/(s (1 + 2 l s))) /; 
  p > 0 && c > 
    p && ((0 < l < p/(2 c) && 
       0 < s < (-2 c + p)/(4 c l - 2 p)) || (p/(2 c) <= l <= 1 && 
       s > 0))

Now the Manipulate can be used (it is a bit slow). I found that l had to be greater than zero and less than 1 in order for the restriction to be satisfied.

Manipulate[
 ParametricPlot[{{p, BRc[p, s, l, c]}, {BRp[c, s, l, p], c}}, {c, 0, 
   50}, {p, 0, 50},
  PlotStyle -> {{Blue, Thick}, {Red, Thick}},
  PlotRange -> {{0, 50}, {0, 50}},
  AspectRatio -> 1,
  PlotLabel -> Style["Best Response Curves", Blue, 20]
  ],

 {{s, 1}, 0, 10, Appearance -> "Open"},
 {{l, 0.001}, 0.001, 1, Appearance -> "Open"}
 ]

I was not able to get AxesLabel to work. My workaround would be to manually label with with text or see if you can find a solution on this site.