An autocatalytic reaction involves a species catalyzing its own formation and can be viewed as a mechanism to explain self-replication. This class of chemical reactions can be used to model phenomena across a wide cross-section of disciplines, i.e chemical reaction engineering, biology, ecology, social sciences and economics. Here, we study a system of two species undergoing autocatalysis. Each species participates in the autocatalysis of the other. The autocatalytic reactions can represent interconversion of two social groups or isomers into each other. The two autocatalytic steps are assumed to be elementary and follow cubic and quadratic laws. We examine the behavior of this coupled autocatalytic system at steady state. Specifically, singularity theory and bifurcation theory are applied to classify the bifurcation behavior of the system. The focus of the work is to obtain the conditions for extinction or complete conversion of one of the species. We analyze the system for all combinations of generation and decay reactions. We find the common features and dominant factors in the different combinations analyzed.