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

On the Convergence of the JKO-scheme and Blow-up of solutions for a Multi-species Chemotaxis System with no Mass Preservation