Graph product representation of organism-environment couplings in evolution

We present a theoretical framework that mathematically formulates the evolutionary dynamics of organism-environment couplings using graph product multilayer networks, i.e., networks obtained by “multiplying” factor networks using some graph product operator. In this framework, one factor network represents different options of environments and their mutual physical reachability, and another factor network represents possible types of organisms and their mutual evolutionary reachability. The organism-environment coupling space is given by a Cartesian product of these two factor networks, and the nodes of the product network represent specific organism-environment combinations. We studied a simple evolutionary model using a reaction-diffusion equation on this organism-environment coupling space. We numerically calculated correlations between the inherent fitness of organisms and the actual average fitness obtained from the graph product-based evolutionary model, varying the spatial diffusion rate while keeping the type diffusion rate small. Results demonstrated that, when the spatial diffusion is sufficiently slow, the correlation between inherent and actual fitnesses drops significantly, where it is no longer valid to assume that fitness can be attributed only to organisms.