(317f) Multiple-Resource Multiple-Species Autocatalysis
Study of autocatalytic reactions has been a popular practice in complex systems science, as a result of the generalizable results of such analysis, which has implications reaching out to broad fields such as population dynamics [1, 2], RNA polymerization, DNA replication [3-5] and stock market dynamics [6-8]. In the past, several studies were carried out using models of cubic autocatalysis in CSTRs. In the case of multiple-species autocatalysis carried out in a CSTR, where at least two autocatalytic species compete for a single resource and none of them are present in the feed stream, it was earlier shown that the coexistence of autocatalysts were globally unstable at steady state [1, 2].
The availability of multiple resources opens up various scenarios that can lead to coexistence of the competing species and has not been studied so far. In this study, we have added another resource to the inlet stream, which can be utilized by either or both autocatalytic species in the CSTR. Bifurcation analyses were carried out using the residence time of the reactor, so as to explore the effect on the stability of coexistence steady states. Three types of scenarios were employed in the analysis, where both species utilize both resources, only one species utilizes both resources, or each species utilizes a single resource.
The results of the bifurcation analyses indicate that the system exhibits a variable multiplicity of 7 to 9 steady states, depending on the residence time of the reactor. It was shown that, while the coexistence steady states are unstable when there are 7 steady states, an additional interaction isola that has a stable zone forms at certain values of the reactor residence time. The robustness of the stable coexistence steady state depends on the specific scenario and rate constants of autocatalysis. This shows that, through careful manipulation of reactor residence time and addition of a secondary resource, stable coexistence of multiple species can be established.
 İ. Birol, F. Teymour, Phys. D Nonlinear Phenom. 144 (2000), 279-297.
 W. Chaivorapoj, İ. Birol, F. Teymour, Ind. Eng. Chem. Res. 41 (2002), 3630-3641.
 L. da Silva, K. C. Mundim, C. Tsallis, Physica A 259 (1998) 415–429.
 P. R. Wills, S. Kauffman, B. M. R. Stadler, P. F. Stadler, Bull. Math. Biol. 60 (1998) 1073–1098.
 P. Schuster, W. Fontana, Physica D 133 (1999) 427–452.
 S. Kauffman, McKinsey Quart. (1995) 118–129.
 C. Blomberg, J. Theoret. Biol. 187 (1997) 541–554.
 J. Hofbauer, J. W. Weibull, J. Econ. Theory 71 (1996) 558–573.