  We consider a degenerate chemotaxis model with two-species and two-stimuli in
dimension $d\geq 3$ and find two critical curves intersecting at one same point
which separate the global existence and blow up of weak solutions to the
problem. More precisely, above these curves (i.e. subcritical case), the
problem admits a global weak solution obtained by the limits of strong
solutions to an approximated system. Based on the second moment of solutions,
initial data are constructed to make sure blow up occurs in finite time below
these curves (i.e. critical and supercritical cases). In addition, the
existence or non-existence of minimizers of free energy functional is discussed
on the critical curves and the solutions exist globally in time if the size of
initial data is small. We also investigate the crossing point between the
critical lines in which a refined criteria in terms of the masses is given
again to distinguish the dichotomy between global existence and blow up. We
also show that the blow ups is simultaneous for both species.
