I thought it would be a good idea to use the command Manipulate so that I could observe the changes in the graph that correspond to changing the values of the constants. To do this, I need the symbolic solution to the system, which I found as follows:
equations = {c'[t] == c[t] (p1*v - (1 - p1) v - dr),
p'[t] == c[t] (1 - p1) v + p[t] (p1*v - (1 - p1) v - dr),
d'[t] == p[t] (1 - p1) v + d[t] (1 - dr)}
DSolve[{equations, c[0] == 1, p[0] == 0, d[0] == 0}, {c[t], d[t],
p[t]}, {t}]
ffc[t_] == E^(t (-dr + (-1 + 2 p1) v))
ffp[t_] ==
1/(-1 - v + 2 p1 v)^2 (-1 + p1)^2 v^2 (E^((1 - dr) t) - E^(
t (-dr + (-1 + 2 p1) v)) - E^(t (-dr + (-1 + 2 p1) v)) t -
E^(t (-dr + (-1 + 2 p1) v)) t v +
2 E^(t (-dr + (-1 + 2 p1) v)) p1 t v)
ffd[t_] == -E^(t (-dr + (-1 + 2 p1) v)) (-1 + p1) t v
total[t_] == ffc[t] + ffp[t] + ffd[t]
Manipulate[
Plot[{ffc[t], ffp[t], ffd[t],total[t]}, {t, 0, 10},
PlotLegends -> {"c[t]", "p[t]", "d[t]", "total cells"}], {{x1, 1,
"Occupany Level at which Division Speed Increases"}, 1,
10}, {{y1, 1, "Division Speed Increase Factor"}, 0,
10}, {{x2, 1,
"Occupancy Level at which Prop. Symmetric Division Increases"}, 1,
10}, {{y2, 1, "Division Proportion Increase Factor"}, 0,
10}, {{v0, 1, "Cell Divisions Per Unit Time"}, 0,
10}, {{p10, .5,
"Proportion of Cells Undergoing Symmetric Division"}, 0,
1}, {{dr, .2, "Cell Death Rate"}, 0,
1}, {{f1, 1, "CSC DLL4 Production"}, 0,
10}, {{f2, 1, "PC DLL4 Production"}, 0,
10}, {{f3, 1, "TDC DLL4 Production"}, 0,
10}, {{r, 1, "Receptors Per Cell"}, 0, 10}, {k, 0, 10}]
This is a very rough attempt and I know that I should include v[t] and p1[t] in the plot too, in addition to several other things, but I can't seem to make the plot show any thing currently. Once I can get it to work I plan on tinkering with it. Anyexplanation as to why nothing shows up?
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