  This work considers a chemotaxis system for multi-species that includes birth
or death rate terms, which implies no mass preservation of the populations. We
aim to show the convergence to a $L^{\infty}\ -\ $weak solutions, that is local
in time, of the JKO - scheme arising from the Optimal Transport Theory, in the
spirit of [35,14]. Currently, $L^{\infty}$ solutions have shown to be important
in order to get uniqueness. Since death rate case does not ensure global
solutions, for arbitrary initial data, in this framework, it could be interest
to analyze the Blowing-up phenomenon of this system. Therefore, in the last
section, we get sufficient conditions that implies blowing-up phenomenon in
finite time and we draw several stages where this occurs. This last part can be
seen as a partial generalization of the blowing-up results in [16].
