  We address the stability of light bullets supported by Bessel optical
lattices with out-of-phase modulation of the linear and nonlinear refractive
indices. We show that spatial modulation of the nonlinearity significantly
modifies the shapes and stability domains of the light bullets. The addressed
bullets can be stable, provided that the peak intensity does not exceed a
critical value. We show that the width of the stability domain in terms of the
propagation constant may be controlled by varying the nonlinearity modulation
depth. In particular, we show that the maximum energy of the stable bullets
grows with increasing nonlinearity modulation depth.
