(583h) Statics and Dynamics of Consecutive and Competitive Cubic Autocatalysis in Well-Mixed Reactors

Autocatalytic reactions and processes are commonly encountered in growth of all living cells, processes involving free radicals, polymerization processes, many inorganic and organic reactions, and crystallization processes. A comprehensive analysis of static and dynamic behavior of series and parallel autocatalytic reactions occurring in a well-mixed reactor is presented. The generation of the autocatalyst from substitutable resources is followed by its decay. Two series or parallel autocatalytic reactions are considered as specific examples. Series and parallel autocatalysis is observed in growth of living cells on multiple substitutable resources (nutrients) and co-metabolism of primary and secondary nutrients (simultaneous utilization of multiple nutrients via independent metabolic pathways inside living cells). A single well-mixed reactor may operate at up to five steady states and can give rise to oscillatory states, including single period and multi-period periodic states, chaos, and aperiodic states. The space of ratio of the kinetic parameters for the autocatalytic reactions and ratio of supply of resources is divided into different regions depending on the maximum number of steady states admissible, which vary from three to five. The space of the remaining kinetic and operating parameters is divided into multiple regions based on the number of physically realizable steady states. This division allows exact determination of appearance and disappearance of particular steady states. Out of the five steady states, instability of two is established analytically. Continuation curves for limit point and Hopf bifurcations are identified in the multi-dimensional kinetic and operating parameter space. The reaction system exhibits a rich variety of steady state patterns. Steady state multiplicity patterns, such as isolas and mushrooms, are predicted in a non-iterative fashion by a judicious choice of parameter combinations. The additional autocatalytic reaction in series or parallel gives rise to much more complex dynamic states over and above what is possible for a single autocatalytic reaction. While the dynamic states are limited to limit cycles in the case of a single autocatalytic reaction, reaction schemes involving two autocatalytic reactions exhibit bifurcations of limit cycles to higher order cyclic states, such as multi-period cyclic states, through period doubling and chaotic behavior. Commensurate with the richness of steady state multiplicity and admissibility of various non-stationary states via Hopf and period-doubling bifurcations, the general reactor dynamics for this reaction scheme consists of interesting and complex patterns.