Mitogen-activated protein kinases (MAPKs) are important signal transducing enzymes, unique to eukaryotes, that are involved in many pathways of cellular regulation. The MAPK cascade is one of the most studied signalling biochemical pathways. This biochemical network involves many proteins, including MAPKs, which communicate by adding phosphate groups to a neighbouring protein, which in turn acts as an "on" or "off" switch. The cascade is well conserved and always consists of a MAPK kinase kinase (MAPKKK), a MAPK kinase (MAPKK), and a MAPK. The work examined the MAPK cascade described by the Ferrell-Huang model. We used methods of the stoichiometric networks analysis and nonlinear dynamics to reduce the network and obtain minimal reaction network for bistability in the MAPK signalling cascade. Finally, we studied properties of the core model for bistability.