We consider the Gierer-Meinhardt system (1.1), shown below, on a bounded smooth domain Ω ⊂ Rn(n ≥ 1) with a homogeneous Neumann boundary condition. For suitable exponents a, b, c and d, we establish certain sufficient conditions for global existence. Theorem 1.1 here, combined with Theorem 1.2 of , implies a classical phenomenon on the effect of the initial data on global existence and finite time blow-up. This work is a continuation of our earlier result  for the Gierer-Meinhardt system. The Gierer-Meinhardt system was introduced in  to model activator-inhibitor systems in pattern formation in ecological systems.