  We study two toy models obtained after a slight modification of the
nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy
models, both consisting of a system of two parabolic equations, we establish
that for data which are, in a suitable sense, smaller than the diffusion
parameter $\tau$ in the equation for the chemoattractant, we obtain global
solutions, and for some data larger than $\tau$ , a finite time blowup. In this
way, we check that our size condition for the global existence is sharp for
large $\tau$ , up to a logarithmic factor.
