Faculty Advisor: Alvin Bayliss and Vladimir Volpert
Curvature & the dynamics of invasion in a continuous-space population model
We consider a partial differential equation (PDE) formulation of the classic and extensively studied Lotka-Volterra model for two species. Individuals of a given species undergo diffusion in two spatial dimensions, and compete with members of the same (resp., rival) species for resources. We consider the consequences of curvature at the interface of regions in which the competing species reside. We show that curvature facilitates competitive exclusion, in which one species displaces the other––to, aside from differences in fitness and mobility, provide a novel and biologically distinct apparatus for invasion in spatially structured environments. The nature, e.g., speed, of invasion is shown to depend predictably on the curvature. Though exclusion does not occur when the interface between the two species is planar, we show that diffusion “smooths” rectangular patches of a given species to facilitate invasion by the surrounding species. Finally, we show that curvature can compensate for differences in fitness and mobility to allow the weaker species to exclude its rival and overtake the landscape. Building on these findings, we aim to (i) consider the regime where intraspecific competition is stronger than interspecific competition and (ii) characterize the stochastic time to invasion upon the introduction of noise into the underlying deterministic model.
Funding: The NSF- Simons Center for Quantitative Biology, Northwestern University