About ghost transients in spatial continuous media

Abstract

The impact of space on ecosystem dynamics has been a matter of debate since the dawn of theoretical ecology. Several studies have revealed that space usually involves an increase in transients’ times, promoting the so-called supertransients. However, the effect of space and diffusion in transients close to bifurcations has not been thoroughly investigated. In non-spatial deterministic models such as those given by ordinary differential equations transients become extremely long in the vicinity of bifurcations. Specifically, for the saddle–node (s–n) bifurcation the time delay, τ, follows τ∼|μ−μc|−1/2; μ and μc being the bifurcation parameter and the bifurcation value, respectively. Such long transients are labeled delayed transitions and are governed by the so-called ghosts. Here, we explore a simple model with intra-specific cooperation (autocatalysis) and habitat loss undergoing a s–n bifurcation using a partial differential equations (PDE) approach. We focus on the effects of diffusion in the ghost extinction transients right after the tipping point found at a critical habitat loss threshold. Our results show that the bifurcation value does not depend on diffusion. Despite transients’ length typically increase close to the bifurcation, we have observed that at extreme values of diffusion, both small and large, extinction times remain long and close to the well-mixed results. However, ghosts lose influence at intermediate diffusion rates, leading to a dramatic reduction of transients’ length. These results, which strongly depend on the initial size of the population, are shown to remain robust for different initial spatial distributions of cooperators. A simple two-patch metapopulation model gathering the main results obtained from the PDEs approach is also introduced and discussed. Finally, we provide analytical results of the passage times and the scaling for the model under study transformed into a normal form. Our findings are discussed within the framework of ecological transients.