  We consider the nonlocal diffusion equation $\partial \_t u=J*u-u+u^{1+p}$ in
the whole of $\R ^N$. We prove that the Fujita exponent dramatically depends on
the behavior of the Fourier transform of the kernel $J$ near the origin, which
is linked to the tails of $J$. In particular, for compactly supported or
exponentially bounded kernels, the Fujita exponent is the same as that of the
nonlinear Heat equation $\partial \_tu=\Delta u+u^{1+p}$. On the other hand,
for kernels with algebraic tails, the Fujita exponent is either of the Heat
type or of some related Fractional type, depending on the finiteness of the
second moment of $J$. As an application of the result in population dynamics
models, we discuss the hair trigger effect for $\partial \_t
u=J*u-u+u^{1+p}(1-u)$
