  We study the critical dynamics of the generalized Smoluchowski-Poisson system
(for self-gravitating Langevin particles) or generalized Keller-Segel model
(for the chemotaxis of bacterial populations). These models [Chavanis & Sire,
PRE, 69, 016116 (2004)] are based on generalized stochastic processes leading
to the Tsallis statistics. The equilibrium states correspond to polytropic
configurations with index $n$ similar to polytropic stars in astrophysics. At
the critical index $n_{3}=d/(d-2)$ (where $d\ge 2$ is the dimension of space),
there exists a critical temperature $\Theta_{c}$ (for a given mass) or a
critical mass $M_{c}$ (for a given temperature). For $\Theta>\Theta_{c}$ or
$M<M_{c}$ the system tends to an incomplete polytrope confined by the box (in a
bounded domain) or evaporates (in an unbounded domain). For $\Theta<\Theta_{c}$
or $M>M_{c}$ the system collapses and forms, in a finite time, a Dirac peak
containing a finite fraction $M_c$ of the total mass surrounded by a halo. This
study extends the critical dynamics of the ordinary Smoluchowski-Poisson system
and Keller-Segel model in $d=2$ corresponding to isothermal configurations with
$n_{3}\to +\infty$. We also stress the analogy between the limiting mass of
white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial
populations in the generalized Keller-Segel model of chemotaxis.
