Optimal dispersal and diffusion-enhanced robustness in two-patch metapopulations: origin’s saddle-source nature matters

A two-patch logistic metapopulation model is investigated both analytically and numerically focusing on the impact of dispersal on population dynamics. First, the dependence of the global dynamics on the stability type of the full extinction equilibrium point is tackled. Then, the behaviour of the total population with respect to the dispersal is studied analytically. Our findings demonstrate that diffusion plays a crucial role in the preservation of both subpopulations and the full metapopulation under the presence of stochastic perturbations. At low diffusion, the origin is a repulsor, causing the orbits to flow nearly parallel to the axes, risking stochastic extinctions. Higher diffusion turns the repeller into a saddle point. Orbits then quickly converge to the saddle’s unstable manifold, reducing extinction chances. This change in the vector field enhances metapopulation robustness. On the other hand, the well-known fact that asymmetric conditions on the patches is beneficial for the total population is further investigated. This phenomenon has been studied in previous works for large enough or small enough values of the dispersal. In this work, we complete the theory for all values of the dispersal. In particular, we derive analytically a formula for the optimal value of the dispersal that maximizes the total population.
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work has been supported by the Spanish State Research Agency (AEI) through Grants PGC2018- 100928-B-I00 (DP), PGC2018-100699-B-I00 (MCIU/AEI/FEDER, UE) (MJC), and by Catalan Government Grant 2017 SGR 1374 (MJC). JS has been supported by the Ramón y Cajal Grant RYC-2017-22243 funded by MCIN/AEI/10.13039/501100011033 “FSE invests in your future”, and by the 2020-2021 Biodiversa+ and Water JPI joint call under the BiodivRestore ERA-NET Cofund (GA No 101003777) pro- ject MPA4Sustainability with funding organizations: Innovation Fund Denmark (IFD), Agence Nationale de la Recherche (ANR), Fundação para a Ciência e a Tecnologia (FCT), Swedish Environmental Protec- tion Agency (SEPA), and grant PCI2022-132926 funded by MCIN/ AEI/10.13039/501100011033 and by the European Union NextGen- erationEU/PRTR. This work has been also funded through the Severo Ochoa and María de Maeztu Program for Centres and Units of Excel- lence in R &D (CEX2020-001084-M). We thank CERCA Programme/ Generalitat de Catalunya for institutional support. DP has been sup- ported by the Research Program 2021–2023 of Universidad Internac- ional de la Rioja. RO thanks to the Faculty for the Future program of the Schlumberger Foundation for the scholarship.
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