About ghost transients in spatial continuous media

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 socalled 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,tau, follows.. tau similar to |mu -mu(c) |(-1/2); mu and mu(c) 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.

Automated summary

The impact of space on ecosystem dynamics, especially in relation to transients and bifurcations, has been the subject of ongoing debate in theoretical ecology. In this study, a simple model with intra-specific cooperation and habitat loss is used to explore the effects of diffusion on ghost extinction transients near a saddle-node bifurcation. The results show that while transients typically increase in length near the bifurcation, at extreme values of diffusion, both small and large, extinction times remain long and similar to well-mixed results. However, at intermediate diffusion rates, the influence of ghosts decreases, leading to a dramatic reduction in transients' length. These findings remain robust for different initial spatial distributions of cooperators, and a two-patch metapopulation model is introduced to summarize the main results. The study also provides analytical results for the model's passage times and scaling in a normal form. The implications of these findings are discussed within the context of ecological transients.