About Ghost Transients in Spatial Continuous Media

16 Pages Posted: 3 Sep 2022

See all articles by Angel Calsina

Angel Calsina

Autonomous University of Barcelona

Silvia Cuadrado

Autonomous University of Barcelona

Blai Vidiella Rocamora

Institut de Biologia Evolutiva (CSIC-Universitat Pompeu Fabra); Centre de Recerca Matemàtica

Josep Sardanyés

Centre de Recerca Matemàtica

Abstract

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

Suggested Citation

Calsina, Angel and Cuadrado, Silvia and Vidiella, Blai and Sardanyés, Josep, About Ghost Transients in Spatial Continuous Media. Available at SSRN: https://ssrn.com/abstract=4208625

Angel Calsina

Autonomous University of Barcelona ( email )

Plaça Cívica
Cerdañola del Valles
Barcelona, 08193
Spain

Silvia Cuadrado

Autonomous University of Barcelona ( email )

Plaça Cívica
Cerdañola del Valles
Barcelona, 08193
Spain

Blai Vidiella

Institut de Biologia Evolutiva (CSIC-Universitat Pompeu Fabra) ( email )

Passeig Marítim de la Barceloneta 37-49
Barcelona, 08003
Spain

Centre de Recerca Matemàtica ( email )

Campus de Bellaterra
Edifici C
Barcelona, 08193
Spain

Josep Sardanyés (Contact Author)

Centre de Recerca Matemàtica ( email )

Campus de Bellaterra
Edifici C
Barcelona, 08193
Spain

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
18
Abstract Views
48
PlumX Metrics