Effective competition determines the global stability of model ecosystems
We investigate the stability of Lotka-Volterra (LV) models constituted by two groups of species such as plants and animals in terms of the intragroup effective competition matrix, which allows separating the equilibrium equations of the two groups. In matrix analysis, the effective competition matrix represents the Schur complement of the species interaction matrix. It has been previously shown that the main eigenvalue of this effective competition matrix strongly influences the structural stability of the model ecosystem. Here, we show that the spectral properties of the effective competition matrix also strongly influence the dynamical stability of the model ecosystem. In particular, a necessary condition for diagonal stability of the full system, which guarantees global stability, is that the effective competition matrix is diagonally stable, which means that intergroup interactions must be weaker than intra-group competition in appropriate units. For mutualistic or competitive interactions, diagonal stability of the effective competition is a sufficient condition for global stability if the inter-group interactions are suitably correlated, in the sense that the biomass that each species provides to (removes from) the other group must be proportional to the biomass that it receives from (is removed by) it. For a non-LV mutualistic system with saturating interactions, we show that the diagonal stability of the corresponding LV system close to the fixed point is a sufficient condition for global stability.
KeywordsEcological models Population dynamics Global stability Structural stability
This work was supported by the Spanish Ministery of Economy through the grant BFU-40020 to UB and FPI grant BES-2009-013072 to APG. Research at the CBMSO is facilitated by the Fundación Ramón Areces. We thank the editor, Sebastian Schreiber, for useful comments that helped to improve the presentation, and an anonymous reviewer for useful suggestions and for finding a typo in a formula in the first version of this paper.
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