Concerning the interactions, smaller variances permit a similar behavior, as illustrated in the insert of Fig 4a. We choose for any non-zero entry of the interaction matrix A in the random model; in the other cases, a strictly positive interaction is randomly drawn from a folded normal distribution such that , and a strictly negative interaction is sampled such that. Concerning local stability within this framework, interestingly, we prove in Proposition S. In other words, any converges to 1 with a variance converging to 0 and is thus trivially positive.
We test now different growth rates. Similar results will be found in S1 Text concerning other types of unstructured models.
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These results rely on the fact that, when the growth rate vector has bounded components, the variance of any goes towards zero. Overall, with weak interactions and for any choice of growth rates, all realisations of the random matrix A lead to a limiting deterministic equilibrium. Thus, abundance variability disappears irrespective of particular realisations of A.
When studying the three other types of unstructured models mutualistic, competitive, and predator-prey networks , we find the same results as above for strong, moderate or weak interactions Table 1. Several works have shown that particular architectures can stabilise ecological networks [ 37 — 40 ]. Nevertheless, the feasibility of different network architectures for different interaction regimes has not been explored. We now test two other food web models that more accurately capture the structure of trophic interactions in real systems, the niche [ 18 ] and the nested-hierarchy models [ 30 ].
The former generates purely interval food webs predators consume all species in a niche interval while the latter is based on evolutionary processes and relaxes this constraint. The results are qualitatively similar to the ones of unstructured random models for all types of interactions. However, two interesting features emerge. First, the mean structural vector now contains negative growth rates, which is the essence of predator-prey systems where predators will die in the absence of prey this feature also appears in the cascade model, but not in the unstructured food web model, see Table 2 in S1 Text.
Second, with weak interactions, each no longer converges to a deterministic limiting vector, but instead to a non-trivial random variable having a distribution of finite support and positive standard-deviation. Thus, contrary to the unstructured and cascade models, a particular realisation of the interaction matrix A will influence the equilibrium values.
Indeed, these highly structured networks and their related mean structural growth rates induce correlations among the abundances at equilibrium, which prevent a convergence to a deterministic value.
We find that the size of the support depends on the hierarchical position of the species Fig 6. We performed additional simulations based on empirical food webs, and find similar results Figs G and H in S1 Text. The species have been assigned a number between zero top species and one basal species corresponding to their hierarchy in the web. The equilibrium converges almost surely to the vector. Since the mean structural vector is constructed for the purpose of maximizing the probability of feasibility, we have also considered random Gaussian growth rates of positive mean to complete our analysis.
For the niche and nested-hierarchy models, the standard deviations of the abundances at equilibrium are ordered according to the trophic position of the species within the food web from basal autotrophs, to intermediate species that are predator and prey, and to top predators.
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We find , a result similar to that obtained with the structural vector, see Figs 3e and 7. This effect, also visible in empirical food webs Figs G and H in S1 Text , disappears in two particular cases: 1 for zero mean Gaussian random growth rates; 2 when the growth rate standard-deviation becomes large. In both cases, the equilibria then become independent of species trophic position Figs G and I in S1 Text. The same parameters as in Fig 6 have been used among simulations.
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The interactions strengths have been simulated similarly as for any structured web. The mean structural vector has been simulated with Monte-Carlo methods trials. For weak interactions in randomly structured systems, feasibility is granted; also, the stability criteria are trivially asymptotically satisfied. Therefore, in such situations, the criteria are basically void of information. However, including realistic structure in trophic systems, we find that the results are no longer trivial. As a general message, future work tackling dynamical stability must be preceded by a feasibility analysis.
In our Lotka-Volterra framework, this involves knowledge of the structure and intensity of interactions, and of the growth rates. Feasibility can then be easily evaluated. Our results indicate that sensible information on the dynamics of natural systems will be obtained by concentrating on the interplay between ecological structure, strengths of interaction weights, and how interaction strengths are distributed in their architecture.
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There are still many other aspects that should be taken into account to reach general conclusions about feasibility and stability that better fit natural systems. They are very different from unstructured systems, and many constraints affect their architecture and dynamics. A first question simply concerns system size in terms of species richness S. Most described systems are middle sized, but large webs do exist [ 41 , 42 ], so that mathematical study focusing on large S limits can be justified.
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An intriguing aspect here is how smaller and well-definable webs are dynamically embedded in larger species-rich systems. For example, it has been found that top predators have a stabilising effect on food webs by coupling fast and slow energy channels in natural aquatic systems [ 43 ].
In this respect, the spatial structure of the system is relevant and has non-trivial consequences for its dynamics [ 45 , 46 ], a research framework that is currently expanding e. Relevant to system size lies also the question of how network nodes are defined: taxonomic resolution is often heterogeneous in observed food webs, with basal species being more often pooled e.
Secondly, species richness S and connectance C are usually treated as independent parameters, while it has been described that C decreases with S in different systems [ 50 ], thus restricting the parameter space to be explored.
Finally, even if topology per se has been suggested to play a minor role in system stability [ 20 ], we showed that network architecture does play a non-trivial role on equilibrium abundances. Layer architectures as those produced by the niche and nested-hierarchy models yield ecologically sensible results; exploring other structural models [ 21 , 51 — 53 ] and especially natural architectures is necessary.
Apart from these structural considerations, ecologists are confronted by a more elusive issue, the estimation of demographic parameters. To make sense, the investigation of feasibility and stability should be based on meaningful parameter values. The growth rate vector r that appears in Lotka-Volterra dynamics is mostly unknown or very difficult to obtain experimentally. One can either keep this vector as an intrinsic free parameter or assume that such growth rates change with interaction weights, as in [ 20 ] or with our mean structural vector.
The estimation of interaction parameters is also far from trivial [ 54 — 57 ]. Experimental studies see, e. Additionally, one unresolved question is how intraspecific competition scales with average interaction strength, which is perhaps the most important ingredient for stability e. This question is related to the study of weak interactions, which have been shown to promote stability [ 19 , 32 , 58 , 59 ] and here feasibility.
Finally, one must consider that these parameters do scale with body size [ 64 ], which defines a constraint between parameter values and network structure, as large-bodied species typically populate higher trophic levels. Interestingly, this allometric relationship may underlie the result that stability depends on the link between trophic position and interaction strength [ 20 ], and our finding that abundance variability scales with trophic position.
For mathematical tractability, many studies on the dynamics of food webs, including ours, rely on Lotka-Volterra models with mass-action type interactions the so-called Holling type I functional response. However, investigations on the number of prey eaten per predator and unit time in simple systems do not support this modelling assumption. There exists a vast literature on the subject and the choice of a sensible functional response is still debated [ 65 , 66 ].
In Section S. It shows that interaction dynamics can strongly affect system feasibility in predator models. Other aspects related to interspecific interactions affect community dynamics, e. Such non-linearities in species interactions have been explored from the stability point of view of systems adjusted to be feasible [ 73 — 75 ]. However, since the precise way interactions are modelled has non-trivial consequences on system dynamics, the relationship between feasibility and complexity should also be considered in such situations.
An articulate answer on the relationship between stability and complexity should obviously incorporate all types of interactions relevant for system dynamics including the effects of ecosystem engineers [ 76 , 77 ]. This question has been the focus of recent theoretical developments [ 78 , 79 ], and empirical studies specifically addressing this question start to emerge [ 80 ]. Science faces increasingly complex situations where high-dimensional parameters occur. A good example is statistical mechanics, for which relevant results have been obtained for highly complex systems without having a precise knowledge of the microscopic details of a model.
Our results like many cited here follow the same line. However, due to the complexity of natural systems, the gap between theoretical and empirical investigations is likely to remain open. To attain a consensus, the feasibility of the model systems should not be forgotten [ 23 , 24 , 26 — 29 ], and more effort must be devoted to obtaining empirical and experimental time series as necessary benchmarks. We thank S.
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