Sustainability without coexistence state in Durrett-Levin hawk-dove model

Models that explain the sustainability of an exploiter-victim ecosystem admit, generally, a coexistence state of both species in the well-mixed limit. Even if this state is unstable, the extinction-prone system may acquire stability on spatial domains where different patches oscillate incoherently around the coexistence state. New experiments, however, suggest that a spatially segregated system may be stable even in the absence of such a coexistence state. Here we revisit the hawk-dove (case 3) model of Durrett and Levin, which has been shown to support persistent population for system of interacting particles. It turns out that this model does not admit a (stable or unstable) coexistence state on a single habitat. We analyze the peculiar mechanism that leads to persistence in this case and the role of demographic stochasticity with and without self-interaction, using numerical simulations and exact solutions in the infinite diffusion limit.