Determination of Bifurcation Points and Catastrophes for the Brusselator Model with Two Parameters

Abstract

In this paper we will present an investigation on some of the features of the Brusselator model. The early origin of this scheme returns to the begin of 1950 when Turing was investigating whether the behaviour of biological systems, in casu an embryological system, could be translated into a mathematical model. He detected that chemical reaction in combination with diffusion, ensuring the transport, are principal mechanisms to explain for example self organization, cell formation, cell division and other phenomena in morphogenetics. The proposed mathematical model was quite complex [1]; 10 reactants were involved together in 8 reaction steps; however it possessed some physical inconveniencies as for example negative concentrations in some cases. Later, Prigogine and his coworkers [2,3] were investigating systems based on nonlinear interactions, because it is possible that such systems can produce highly symmetric structures due to their self-organizing capacities. They claimed that a better and functional organization of a reacting system increases the intensity of the related process: in chemical reaction systems for example reaction rates increase. Prigogine and Nicolis compare herefore the non-equilibrium system with a company, where a spacial or temporal reorganization of the work leads to a higher efficiency.