If I understand it correctly, I think the following is what you want:
Solve the differential equation of n variables, with initial conditions defined using the previous differential equation solution of n-1 variables and with an initial condition for the last variable (which randomly depends on one of the previous variables).
The following code uses iniCond which generates initial conditions for a given i, diffGen generates equations with the initial conditions, and finally solveDiffGen solves the differential equations.
T = 50;
nu = 0.05;
vars[i_] := Subscript[x, #][t] & /@ Range[i];
iniCond[1] = {Subscript[x, 1][0] == 0.7};
diffGen[i_] :=
Join[Table[{Subscript[x, j]'[t] ==
Subscript[x, j][
t] (1 - Subscript[x, j][t] -
nu (Sum[Subscript[x, k][t] Boole[k != j], {k, i}]))}, {j,
i}], iniCond[i]] // Flatten;
iniCond[i_] :=
Module[{prevSolConds, lastCond},
prevSolConds =
NDSolve[diffGen[i - 1],
vars[i - 1], {t, 0,
T}] /. (x_ -> g_) :> {(x /. {t -> 0}) == (g /. {t -> T})} //
Flatten;
lastCond =
Subscript[x, i][0] ==
0.01 Subscript[x, RandomInteger[{1, i - 1}]][t] /. t -> T;
{prevSolConds, lastCond} // Flatten];
solveDiffGen[i_] := NDSolve[diffGen[i], vars[i], {t, 0, T}];