Group methods applied to a reaction-diffusion system generalizing Proteus Mirabilis models

•We write a reaction diffusion system that can reduce to that proposed by Medvedev, Kaper, Kopell (MKK).•The system contains as a special case also the model proposed by Czirok, Matsushita and Vicsek (CMV).•Some equivalence transformation algebras for the class of system to whom belongs the model are determined.•From a special equivalence algebra we obtain some constitutive function forms that extend the principal Lie Algebra.•For some constitutive functions of models MKK and CMV, symmetry reductions and new exact invariant solutions are obtained. Starting from a well-known model introduced by Medvedev, Koper and Kopell (MKK) we have written a reaction-diffusion system, that models the evolution of Proteus Mirabilis bacterial colonies, and have considered the class of pde systems to whom it belongs. Once specialized some parameters we are able to get both the MKK models and the models introduced by Czirok, Matsushita and Vicsek (CMV). After having derived some equivalence algebras of this class, we use them to find forms of constitutive functions that allow to extend the principal Lie algebra. For some constitutive functions of models MKK and CMV, symmetry reductions and new exact invariant solutions are obtained.