Autocatalytic reactions are elementary and non-elementary reactions where a molecular species serves as both reactant and product and is labeled as an auto-catalyst. There is net generation of auto-catalyst as a result of the reaction. Energy is an auto-catalyst for exothermic reactions as higher temperature (higher energy) implies higher reaction rate, which implies higher energy generation rate, which leads to higher temperature. Energy generated therefore catalyzes exothermic reactions. A few examples of non-reactive autocatalytic processes are crystallization, precipitation, and ice formation. Autocatalytic reactions are ubiquitous in sustenance of life on this planet. Cells in a variety of living species and organisms consume nutrients (reactants) to produce more cells. Cells as such serve both as a reactant and product, with net production of cells leading to cell growth. We consider here population balances for and dynamics of autocatalytic replicators, with the kinetics of replication being proportional to concentration of the auto-catalyst (quadratic autocatalysis) or square of the same (cubic autocatalysis). For the sake of illustration, we consider a single auto-catalyst or two auto-catalysts. The kinetic parameters associated with resource utilization, auto-catalyst replication, and yield of auto-catalyst are considered to be variable parameters. The population dynamics reveals rich patterns of pathological behavior as the kinetic parameters are varied. These patterns range from fixed points to cyclic behavior with odd and even number of periods to aperiodic behavior and chaotic behavior. Transition from fixed points to cyclic behavior occurs upon loss of stability of fixed points. For both cycles with even number of periods and odd number of periods, loss of stability leads to cycles of higher number of periods through period doubling. There are bursts of aperiodic behavior and chaotic behavior in certain parameter ranges. Cycles of period 2^n, n = 0 and higher, are easier to tract and cycles of period (2m+1)*2^n, n = 0 and higher and m = 1 and higher, are much more difficult to tract. Stability characteristics of fixed points and cycles with finite number of periods are examined systematically and in detail. Numerical results reveal the qualitative differences between quadratic and cubic autocatalysis. The pathological behavior gets richer with the addition of another autocatalytic species, the interaction between the two autocatalytic species being confined to be competitive or cooperative. Numerical results reveal the rich diversity of behavior of single auto-catalyst subject to quadratic and cubic autocatalysis and interesting features of two interacting autocatalytic species.
