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. (C) 2018 Elsevier B.V. All rights reserved.