Influence of External Fluctuations on a Hypercycle Formed by Two Kinetically Indistinguishable Species

The effects of a fluctuating environment on a model that describes the behavior of self-replicative molecules forming a hypercycle organization is analyzed. We assume that the mutation rate, or probability of erroneous self-replication of the species, 1-Q, is affected by a random external noise, and in order to make a more realistic biological model, we consider a level of extinction for each species. The qualitative theory of stochastic dynamical systems is used to find the asymptotic behavior of the model depending on the range of variation of Qt, and the possibility for the system of maintaining the hypercyclic organization under the influence of the stochastic fluctuations is discussed. A new stochastic bifurcation diagram is drawn as a function of the mean value 〈Qt〉, and numerical simulations demonstrate the main conclusions obtained from the theoretical analysis.