  The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles
moving in $R^d$ according to a symmetric $\alpha$-stable L\'evy process
$(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law
$(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random measure
with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. By
means of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$,
occupation time fluctuation limits for the system as $T\to\infty$ have been
obtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneous
case), and finite measures $(\gamma>d)$. In some cases $H_T\equiv 1$ and in
others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit
processes are quite different for Lebesgue and for finite measures. Therefore
the question arises of what kinds of limits can be obtained for Poisson
intensity measures that are intermediate between Lebesgue measure and finite
measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used
for investigating this question. Occupation time fluctuation limits are
obtained which interpolate in some way between the two previous extreme cases.
The limit processes depend on different arrangements of the parameters
$d,\alpha,\beta,\gamma$. Related results for the corresponding
$(d,\alpha,\beta,\gamma)$-superprocess are also given.
