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 the previous variable).

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.

    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[{conditions},
       conditions = 
        NDSolve[diffGen[i - 1], 
           vars[[i - 1]], {t, 0, 
            T}] /. (x_ -> g_) :> {(x /. {t -> 0}) -> (g /. {t -> T})} // 
         Flatten;
       {conditions /. 
          Rule -> Equal, {Subscript[x, i][0] == 
            0.01 Subscript[x, RandomInteger[{1, i - 1}]][t] /. t -> T}} //
         Flatten
       ];
    solveDiffGen[i_] := NDSolve[diffGen[i], vars[[i]], {t, 0, T}];