About Ghost Transients in Spatial Continuous Media
16 Pages Posted: 3 Sep 2022
The impact of space in ecosystems' dynamics has been a matter of debate in the last decades. Several models have revealed that space typically involves longer transients (so-called super transients). However, the effect of space and diffusion in transients close to bifurcations has not been thoroughly investigated. Non-spatial deterministic models, such as those given by ordinary differential equations, have revealed that transients become very long close to bifurcations. Specifically, for the saddle-node (s-n) bifurcation the time delay, [[EQUATION]] , follows [[EQUATION]] ; [[EQUATION]] and [[EQUATION]] 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 habitat loss threshold given by a s-n bifurcation. We 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. 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 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.
Keywords: Reaction-diffusion dynamics, Saddle-node bifurcations, Scaling laws, Spatial ecology, Tipping points, Transients
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