The system has several nonlinearities, which might make it difficult/impossible to find an analytical solution. I think that there was a misinterpretation of the input. In your first post you apparently tried to assign values to the parameters, but you used double equals which indicate an equation rather than an assignment.
a == 15.5, b == 1.3710^-4, c == 1.05 * 10^8, e == 1000
In that case Mathematica does (rightly) not find a solution. Also, you might want to run this:
ClearAll["Global`*"]
sol = ParametricNDSolveValue[{
m'[t] == -2 a (m[t]^2) + 2 b*d[t] - c*m[t] d[t] + e*o[t],
d'[t] == a (m[t]^2) - b*d[t] - c*m[t] d[t] + e*o[t],
o'[t] == c*m[t] d[t] - e*o[t], m[0] == 1, d[0] == 0, o[0] == 0},
{m, d, o}, {t, 0, 100}, {a, b, c, e}]
Manipulate[x = sol[a, b, c, e];
LogPlot[{x[[1]][t], x[[2]][t], x[[3]][t]}, {t, 0, 100}], {{a,
15.5}, -500, 500}, {{b, 0.0001371}, -1, 1}, {{c, 1.05*10^8}, 10^2,
10^10}, {{e, 1000}, -1000, 10000}]
You can modify the values of the parameters and see how your solution changes. Note that for some combinations of parameter values the solution diverges or leads to numerical instabilities.