Stability criteria for complex ecosystems

Abstract

Forty years ago, May proved that sufficiently large or complex ecological networks have a probability of persisting that is close to zero, contrary to previous expectations. May analysed large networks in which species interact at random. However, in natural systems pairs of species have well-defined interactions (for example predator-prey, mutualistic or competitive). Here we extend May's results to these relationships and find remarkable differences between predator-prey interactions, which are stabilizing, and mutualistic and competitive interactions, which are destabilizing. We provide analytic stability criteria for all cases. We use the criteria to prove that, counterintuitively, the probability of stability for predator-prey networks decreases when a realistic food web structure is imposed or if there is a large preponderance of weak interactions. Similarly, stability is negatively affected by nestedness in bipartite mutualistic networks. These results are found by separating the contribution of network structure and interaction strengths to stability. Stable predator-prey networks can be arbitrarily large and complex, provided that predator-prey pairs are tightly coupled. The stability criteria are widely applicable, because they hold for any system of differential equations.

The ghost of nestedness in ecological networks

“…The lack of nestedness in the dominant quantitative structure of empirical networks is consistent with our mathematical treatment of the local stability of nested structures. Although local stability analysis captures only one aspect of ecological system dynamics, its mathematical tractability provides a good starting point for assessing the dynamical consequences of network structure 25 .…”

“…If all of the signs are negative then the equilibrium point is stable. Contemporary work has shown that nested mutualistic networks are less likely to be stable than their random counterparts 25 . We now demonstrate that a nested structure within M minimizes local stability.…”

“…For very large systems, the spectral radius of M is rðMÞ % rðM 0 Þ À D, where À D is the average value of the diagonal 25 . Stability is therefore achieved whenever rðM 0 Þ o D. Analogous to arranging coefficients in an adjacency or incidence matrix (as we did above), among all possible ways of arranging the coefficients of M 0 the configuration yielding the largest spectral radius is perfect nestedness (binary or quantitative).…”

Emergence of structural and dynamical properties of ecological mutualistic networks

“…Nevertheless, the degree of nestedness varies among networks. Recently 5,10 , it has been argued that nestedness increases biodiversity and begets stability, but these results are in conflict with robust theoretical evidences showing that ecological communities with nested interactions are inherently less stable than unstructured ones 12,14,15 and that mutualism could be detrimental to persistence 11,15 . We aim to elucidate general optimization mechanisms underlying network structure and its influence on community dynamics and stability.…”

“…The relatively low abundances of the specialist species make them more vulnerable to extinction and results in correspondingly lower community resilience, as measured by the maximum real part of the eigenvalues of the community matrix community ( Figure 4B). The advantage of having a high total population leading to increased robustness against extinction due to demographic fluctuations, carries with it the cost associated with a lower resilience -the optimized network recovers from perturbations on a longer timescale than its random counterpart 12,14,27 (Figure 4C). …”