Chaotic Stability in Spatially-Resolved Host-parasite Replicators: the Red Queen on a Lattice

Evolutionary patterns of change are often linked to complex coupled responses associated to predator-prey, immune system-viruses or host–parasite interactions. Under some conditions, it has been postulated that the final outcome of such interactions are oscillatory patterns of genotypic and phenotypic change, generically labeled as Red Queen dynamics (RQD). In RQD, changes occur in a changing pattern ensuring the persistence of both partners at the cost of constant changing. In this paper, we analyze the dynamics of two populations of host-parasite replicators extended on a surface considering three scenarios associated to neutral landscapes on binary hypercubes. Stochastic simulations show three asymptotic dynamic states: (i) host survival and parasite extinction, (ii) host-parasite extinction, and (iii) a stable RQD scenario able to maintain chaotic oscillations. The RQD scenario is shown to be facilitated with the increase of the genotypic diversity associated to the increase of the length of the replicators. Such diversity, jointly with the inclusion of spatial degrees of freedom andmutation, can act as a stabilizing factor.