In order to analytically define the stochastic model for the symmetric hypercycle we need to consider both replication and diffusion processes. This means to either explore a continuous, Turing-like approach or instead properly define the discrete rules of interaction. Although the introduction of diffusion might be a difficult task (Fontanari and Ferreira, 2002), here we show that such a model can be derived from a metapopulation approach (Bascompte and Solé, 1998 and Hanski, 1999). Actually, as far as we know, this is the first time an explicit discrete model allows a mean field treatment of diffusion on a metapopulation context.
where e and r represent decay and replication rates, respectively; D1 and D-1 are the diffusion rates attending a bidirectional diffusion process (we assume that D1=D-1≡D).
Space has played a key role in canalizing the evolution and selection of early replicators and is likely to be important in future, experimental synthesis of self-replicating molecules. Understanding the nature of transitions between different behavioral patterns is a relevant step in defining the potential scenarios where these replicators might successfully persist and evolve. In the present work we develop several models in order to analyse the dynamics of two-member hypercycles. The continuous, non-spatial dynamical system shows the existence of a critical decay rate below which three fixed points are shown to exist: two stable points and a saddle-node. The stable points are the attractors involved in both persistence (i) and extinction (ii) phases. Beyond εc and after the coalescence of the coexistence node with the saddle-node in a saddle-node bifurcation that leaves a ghost in the phase plane, the only attractor of phase space is the extinction one, thus both hypercycle members asymptotically extinct.
The mean field model predicts the hyperbolic relation between diffusion associated with cross-catalytic replication (labeled δ) and replication. The spatial analysis developed with the stochastic CAs allows us to take into account the local nature of interactions. We also found extinction and coexistence phases, which are separated by sharp boundaries. Such models show that diffusion plays a positive role in the persistence of both replicators. In the two-dimensional CA the extinction phase is shown to be given by an absorbing state associated with a first-order phase transition, analogous to the saddle-node bifurcation scenarios shown in the mean field approaches. We have shown that the two-member hypercycle as a surface-bonded chemical network could increase its persistence in a medium favoring diffusion. Nevertheless, diffusion might play a negative role when considering extremely low densities of replicators as initial conditions. Finally, a novel mean field metapopulation model considering diffusion (D) of replicators has also been considered. The qualitative change between phases (i) and (ii) is shown and is also characterized by a saddle-node bifurcation.
Our results confirm the relevance of spatial degrees of freedom as well as of the information encoded by the hypercycle in shaping the dynamics of hypercyclic organization. As it occurs with other metapopulation models, we also show the importance of dispersal (diffusion) as a key for success. Under similar conditions, the presence of diffusion allows the expansion through available space and the increasing opportunities for colonization increase the chances of maintaining a stable population. Future work should consider including in our description the presence of adaptive dynamics, thus allowing the offspring of the hypercycles to explore parameter space. In this context, the emergence of parasites, the role of stochasticity and the relevance of spatial structures in hypercyclic evolution should be analysed in more detail.