Spatio-temporal Moran dynamics in continuous media

Abstract: Understanding how natural selection unfolds across space and time is a central problem in evolutionary biology. Classic models such as the Moran process capture stochastic birth-death dynamics in structured populations, while reaction-diffusion equations like the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation describe deterministic wave-like spread. In this work, we bridge these perspectives by deriving partial differential equations for the spatiotemporal limit of Moran dynamics in continuous media. Our model incorporates two distinct fitness components: fecundity (birth rate) and viability (death rate). We demonstrate that the resulting selective wave speeds differ substantially in spatial Moran Birth-death (Bd), Moran Death-birth (Db), and FKPP dynamics. When fecundity drives the dynamics, we observe that the selective waves decelerate for the Bd process, whereas in the Db process the wave propagates with a higher, constant speed. In contrast, when viability drives the process, the Db wave accelerates, while the Bd and FKPP waves maintain comparable constant speeds. We extend the framework to heterogeneous media, represented as weighted lattice graphs in one or two dimensions. We derive a continuous space analog of isothermal graphs and establish that the isothermality condition corresponds to the conservation of a local current.