How the Dimension of Space Affects the Products of Pre-Biotic Evolution: The Spatial Population Dynamics of Structural Complexity and The Emergence of Membranes

We show that autocatalytic networks of ε-machines and their population dynamics differ substantially between spatial (geographically distributed) and nonspatial (panmixia) populations. Generally, regions of spacetime-invariant autocatalytic networks--or domains--emerge in geographically distributed populations. These are separated by functional membranes of complementary ε-machines that actively translate between the domains and are responsible for their growth and stability. We analyze both spatial and nonspatial populations, determining the algebraic properties of the autocatalytic networks that allow for space to affect the dynamics and so generate autocatalytic domains and membranes. In addition, we analyze populations of intermediate spatial architecture, delineating the thresholds at which spatial memory (information storage) begins to determine the character of the emergent autocatalytic organization.