Adding labels to simplex and removing extra axes

Question

I am trying to remove certain parts from my 3d-plots and also add labels to the vertices. In the attached plot, I would only like to keep the 3d-simplex with all the blue lines. I would like to remove the extra axes. In addition, I would like to add labels 'X', 'Y' and 'Z' to the vertices of the simplex. Any help will be greatly appreciated.

The code has two plots, one plot is the simplex joining the points (1,0,0), (0,1,0) and (0,0,1) and the second plot is the trajectories of certain quantities which am interested in. The three quantities of my interest always sump up to 1 and hence lie on the simplex at all times. Here is the code am using:

Clear[f, g, h, p, r, l, jac, u1, u2, u3, u4, u5, G, plotfunc0]
r = 0.3;  
g = 0.5;
b = 0.1/10;  
c = 0.2/10;
G = {{0, 1 + g + b + c, b + c, 1 + g, b, 1 + g + b}, {1 - g - b - c, 
    1, 1 - g, 1 - b - c, 1 - g - c, 1 - c}, {-b - c, 1 + g, 0, 
    1 + g - b - c, -c, 1 + g - c}, {1 - g, 1 + b + c, 1 - g + b + c, 
    1, 1 - g + b, 1 + b}, {-b, 1 + g + c, c, 1 + g - b, 0, 
    1 + g}, {1 - g - b, 1 + c, 1 - g + c, 1 - b, 1 - g, 1}};
u1[hH_, dH_, hD_, dD_, hV_] = 
  G[[1, 1]]*hH +  G[[1, 2]]*dH + G[[1, 3]]*hD +  G[[1, 4]]*dD +  
   G[[1, 5]]*hV +  G[[1, 6]]*(1 - hH - dH - hD - dD - hV);
u2[hH_, dH_, hD_, dD_, hV_] = 
 G[[2, 1]]*hH +  G[[2, 2]]*dH + G[[2, 3]]*hD +  G[[2, 4]]*dD +  
  G[[2, 5]]*hV +  G[[2, 6]]*(1 - hH - dH - hD - dD - hV); 
u3[hH_, dH_, hD_, dD_, hV_] = 
 G[[3, 1]]*hH +  G[[3, 2]]*dH + G[[3, 3]]*hD +  G[[3, 4]]*dD +  
  G[[3, 5]]*hV +  G[[3, 6]]*(1 - hH - dH - hD - dD - hV);
u4[hH_, dH_, hD_, dD_, hV_] = 
  G[[4, 1]]*hH +  G[[4, 2]]*dH + G[[4, 3]]*hD +  G[[4, 4]]*dD +  
   G[[4, 5]]*hV +  G[[4, 6]]*(1 - hH - dH - hD - dD - hV);
u5[hH_, dH_, hD_, dD_, hV_] = 
  G[[5, 1]]*hH +  G[[5, 2]]*dH + G[[5, 3]]*hD +  G[[5, 4]]*dD +  
   G[[5, 5]]*hV +  G[[5, 6]]*(1 - hH - dH - hD - dD - hV);
u6[hH_, dH_, hD_, dD_, hV_] = 
  G[[6, 1]]*hH +  G[[6, 2]]*dH + G[[6, 3]]*hD +  G[[6, 4]]*dD +  
   G[[6, 5]]*hV +  G[[6, 6]]*(1 - hH - dH - hD - dD - hV);
ualpha[hH_, dH_, hD_, dD_, 
   hV_] = (hH*u1[hH, dH, hD, dD, hV]) + (dH*
     u2[hH, dH, hD, dD, hV]) + (hD*u3[hH, dH, hD, dD, hV]) + (dD*
     u4[hH, dH, hD, dD, hV]) + (hV*
     u5[hH, dH, hD, dD, hV]) + ((1 - hH - dH - hD - dD - hV)*
     u6[hH, dH, hD, dD, hV]);
uh[hH_, dH_, hD_, dD_, 
   hV_] = (hH*u1[hH, dH, hD, dD, hV]) + (hD*
     u3[hH, dH, hD, dD, hV]) + (hV*u5[hH, dH, hD, dD, hV]);
ud[hH_, dH_, hD_, dD_, 
   hV_] = (dH*u2[hH, dH, hD, dD, hV]) + (dD*
     u4[hH, dH, hD, dD, hV]) + ((1 - hH - dH - hD - dD - hV)*
     u6[hH, dH, hD, dD, hV]);
uH[hH_, dH_, hD_, dD_, 
   hV_] = (hH*u1[hH, dH, hD, dD, hV]) + (dH*u2[hH, dH, hD, dD, hV]);
uD[hH_, dH_, hD_, dD_, 
   hV_] = (hD*u3[hH, dH, hD, dD, hV]) + (dD*u4[hH, dH, hD, dD, hV]);
uV[hH_, dH_, hD_, dD_, 
   hV_] = (hV*u5[hH, dH, hD, dD, hV]) + ((1 - hH - dH - hD - dD - hV)*
     u6[hH, dH, hD, dD, hV]);
F1[hH_, dH_, hD_, dD_, 
   hV_] = ((1 - r)*hH*
     u1[hH, dH, hD, dD, hV]/ualpha[hH, dH, hD, dD, hV]) + (r*
     uh[hH, dH, hD, dD, hV]*
     uH[hH, dH, hD, dD, hV]/((ualpha[hH, dH, hD, dD, hV])^2)) - hH;
F2[hH_, dH_, hD_, dD_, 
   hV_] = ((1 - r)*dH*
     u2[hH, dH, hD, dD, hV]/ualpha[hH, dH, hD, dD, hV]) + (r*
     ud[hH, dH, hD, dD, hV]*
     uH[hH, dH, hD, dD, hV]/((ualpha[hH, dH, hD, dD, hV])^2)) - dH;
F3[hH_, dH_, hD_, dD_, 
   hV_] = ((1 - r)*hD*
     u3[hH, dH, hD, dD, hV]/ualpha[hH, dH, hD, dD, hV]) + (r*
     uh[hH, dH, hD, dD, hV]*
     uD[hH, dH, hD, dD, hV]/((ualpha[hH, dH, hD, dD, hV])^2)) - hD;
F4[hH_, dH_, hD_, dD_, 
   hV_] = ((1 - r)*dD*
     u4[hH, dH, hD, dD, hV]/ualpha[hH, dH, hD, dD, hV]) + (r*
     ud[hH, dH, hD, dD, hV]*
     uD[hH, dH, hD, dD, hV]/((ualpha[hH, dH, hD, dD, hV])^2)) - dD;
F5[hH_, dH_, hD_, dD_, 
   hV_] = ((1 - r)*hV*
     u5[hH, dH, hD, dD, hV]/ualpha[hH, dH, hD, dD, hV]) + (r*
     uh[hH, dH, hD, dD, hV]*
     uV[hH, dH, hD, dD, hV]/((ualpha[hH, dH, hD, dD, hV])^2)) - hV;
nmax = 100.0;
tmax = 1000000;
func0 = {};
P0 = {};
plotfunc0 = {};
For[k = 1, k <= nmax, k++, 
 region = ImplicitRegion[
   hH + dH + hD + dD + hV <= 1 && hH >= 0 && dH >= 0 && hD >= 0 && 
    dD >= 0 && hV >= 0, {hH, dH, hD, dD, 
    hV}];(*way to get uniform points from region*)
 rand = RandomPoint[region] ;
 solution = 
  NDSolve[{hH'[t] == F1[hH[t], dH[t], hD[t], dD[t], hV[t]], 
    dH'[t] == F2[hH[t], dH[t], hD[t], dD[t], hV[t]], 
    hD'[t] == F3[hH[t], dH[t], hD[t], dD[t], hV[t]], 
    dD'[t] == F4[hH[t], dH[t], hD[t], dD[t], hV[t]], 
    hV'[t] == F5[hH[t], dH[t], hD[t], dD[t], hV[t]], 
    hH[0] == rand[[1]], dH[0] == rand[[2]], hD[0] == rand[[3]], 
    dD[0] == rand[[4]], hV[0] == rand[[5]]}, {hH, dH, hD, dD, hV}, {t,
     0, tmax}];
 plotfunc0 = 
  ParametricPlot3D[{hH[t] + dH[t], hD[t] + dD[t], 
      1 - hH[t] - dH[t] - hD[t] - dD[t]} /. solution, {t, 0, tmax}, 
    BaseStyle -> Arrowheads[{0, .01, 0.01, 0}], 
    PlotStyle -> {Blue, Thin}, Boxed -> False, 
    PlotRange -> {{0, 1}, {0, 1}, {0, 1}}] /. Line -> Arrow;
 AppendTo[func0, plotfunc0]]
p1 = Graphics3D[Simplex[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}], 
   Boxed -> False];
Show[func0, p1]

Answer 1

A bit more elementary than kglr's comment, and with the caveat that it is not clear what label you want to place where, replace your last two commands with

p1 = Graphics3D[{Simplex[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}],
    Text["X", {.55, -.1, .55}],
    Text["Y", {-.1, .55, .55}],
    Text["Z", {.55, .55, -.1}]}, Boxed -> False];
Show[func0, p1, Axes -> False]

to get something like