The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate

When can higher-order interactions produce stable coexistence?

Most ecological models are based on the assumption that species interact in pairs. Diverse communities, however, can have higher-order interactions, in which two or more species jointly impact the growth of a third species. A pitfall of the common pairwise approach is that it misses the higher-order interactions potentially responsible for maintaining natural diversity. Here, we explore the stability properties of systems where higher-order interactions guarantee that a specified set of abundances is a feasible equilibrium of the dynamics. Even these higher-order interactions which lead to equilibria do not necessarily produce stable coexistence. Instead, these systems are more likely to be stable when the pairwise interactions are weak or facilitative. Correlations between the pairwise and higher-order interactions, however, do permit robust coexistence even in diverse systems. Our work not only reveals the challenges in generating stable coexistence through higher-order interactions but also uncovers interaction patterns that can enable diversity.

Food-web complexity, consumer behavior, and diet specialism: impacts on ecosystem stability

Ecological stability is a fundamental aspect of food web dynamics. In this study, we explore the factors influencing stability in complex ecological networks, characterizing it through biomass oscillations and species persistence. Using an Extended Niche model, we generate diverse food web structures and investigate the effects of intraspecific consumer interference, network size, connectance, and diet specialism on stability. Our findings reveal that intraspecific consumer interference plays a pivotal role in shaping stability. Higher interference results in stable dynamics, reducing oscillations and extinctions. Additionally, differences emerge between food webs comprised of invertebrate consumers and those of ectotherm vertebrates, with the latter showing higher oscillations. Network size and connectance also influence stability, where larger and more connected webs tend to exhibit reduced oscillations. Overall, our study sheds light on the complex interplay of factors affecting ecological stability in food webs. Understanding these dynamics is crucial for biodiversity conservation and ecosystem management.

Supplementary Information

The online version contains supplementary material available at 10.1007/s12080-024-00580-w.

Unlocking ensemble ecosystem modelling for large and complex networks

The potential effects of conservation actions on threatened species can be predicted using ensemble ecosystem models by forecasting populations with and without intervention. These model ensembles commonly assume stable coexistence of species in the absence of available data. However, existing ensemble-generation methods become computationally inefficient as the size of the ecosystem network increases, preventing larger networks from being studied. We present a novel sequential Monte Carlo sampling approach for ensemble generation that is orders of magnitude faster than existing approaches. We demonstrate that the methods produce equivalent parameter inferences, model predictions, and tightly constrained parameter combinations using a novel sensitivity analysis method. For one case study, we demonstrate a speed-up from 108 days to 6 hours, while maintaining equivalent ensembles. Additionally, we demonstrate how to identify the parameter combinations that strongly drive feasibility and stability, drawing ecological insight from the ensembles. Now, for the first time, larger and more realistic networks can be practically simulated and analysed.

Stochastic model corrections for reduced Lotka-Volterra models exhibiting mutual, commensal, competitive, and predatory interactions

We explore model-form error and how to correct it in systems of ordinary differential equations. In particular, we focus on the Lotka-Volterra equations, which are used broadly in fields such as ecology, biology, economics, chemistry, and physics. Accounting for every object and their complex interactions with a complete model often becomes infeasible, thereby requiring reduced models. However, reduced models may omit vital relationships, resulting in discrepancies between reduced model predictions and observations from the true system. In this work, we propose a model correction framework for decreasing such discrepancies. Specifically, we embed a stochastic enrichment operator into the reduced model's system of equations. The enrichment operator is theory-informed, calibrated with observations from the complete model, and extended to extrapolative combinations of parameters and initial conditions. The complete model involves N species, while the reduced and enriched models only track M Collapse

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Affiliation(s)

R Bandy Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80309, USA
R Morrison Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80309, USA

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Structural asymmetry in biotic interactions as a tool to understand and predict ecological persistence

A universal feature of ecological systems is that species do not interact with others with the same sign and strength. Yet, the consequences of this asymmetry in biotic interactions for the short- and long-term persistence of individual species and entire communities remains unclear. Here, we develop a set of metrics to evaluate how asymmetric interactions among species translate to asymmetries in their individual vulnerability to extinction under changing environmental conditions. These metrics, which solve previous limitations of how to independently quantify the size from the shape of the so-called feasibility domain, provide rigorous advances to understand simultaneously why some species and communities present more opportunities to persist than others. We further demonstrate that our shape-related metrics are useful to predict short-term changes in species' relative abundances during 7 years in a Mediterranean grassland. Our approach is designed to be applied to any ecological system regardless of the number of species and type of interactions. With it, we show that is possible to obtain both mechanistic and predictive information on ecological persistence for individual species and entire communities, paving the way for a stronger integration of theoretical and empirical research.

Equilibrium and surviving species in a large Lotka-Volterra system of differential equations

Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving species. This heuristics combines arguments from Random Matrix Theory, mathematical optimization (LCP), and standard extreme value theory. Numerical simulations, together with an empirical study where the strength of interactions evolves with time, illustrate the accuracy and scope of the results.

Feasibility and stability in large Lotka Volterra systems with interaction structure

Complex system stability can be studied via linear stability analysis using random matrix theory (RMT) or via feasibility (requiring positive equilibrium abundances). Both approaches highlight the importance of interaction structure. Here we show, analytically and numerically, how RMT and feasibility approaches can be complementary. In generalized Lotka-Volterra (GLV) models with random interaction matrices, feasibility increases when predator-prey interactions increase; increasing competition/mutualism has the opposite effect. These changes have crucial impact on the stability of the GLV model.

Non-random interactions within and across guilds shape the potential to coexist in multi-trophic ecological communities

Theory posits that the persistence of species in ecological communities is shaped by their interactions within and across trophic guilds. However, we lack empirical evaluations of how the structure, strength and sign of biotic interactions drive the potential to coexist in diverse multi-trophic communities. Here, we model community feasibility domains, a theoretically informed measure of multi-species coexistence probability, from grassland communities comprising more than 45 species on average from three trophic guilds (plants, pollinators and herbivores). Contrary to our hypothesis, increasing community complexity, measured either as the number of guilds or community richness, did not decrease community feasibility. Rather, we observed that high degrees of species self-regulation and niche partitioning allow for maintaining larger levels of community feasibility and higher species persistence in more diverse communities. Our results show that biotic interactions within and across guilds are not random in nature and both structures significantly contribute to maintaining multi-trophic diversity.

Maximal ecological diversity exceeds evolutionary diversity in model ecosystems

Understanding community saturation is fundamental to ecological theory. While investigations of the diversity of evolutionary stable states (ESSs) are widespread, the diversity of communities that have yet to reach an evolutionary endpoint is poorly understood. We use Lotka-Volterra dynamics and trait-based competition to compare the diversity of randomly assembled communities to the diversity of the ESS. We show that, with a large enough founding diversity (whether assembled at once or through sequential invasions), the number of long-time surviving species exceeds that of the ESS. However, the excessive founding diversity required to assemble a saturated community increases rapidly with the dimension of phenotype space. Additionally, traits present in communities resulting from random assembly are more clustered in phenotype space compared to random, although still markedly less ordered than the ESS. By combining theories of random assembly and ESSs we bring a new viewpoint to both the saturation and random assembly literature.

Feasibility of sparse large Lotka-Volterra ecosystems

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with [Formula: see text] other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n. We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d, namely [Formula: see text] or [Formula: see text] with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution [Formula: see text] (component-wise) to the equilibrium linear equation: [Formula: see text]where [Formula: see text] is the [Formula: see text] vector with components 1 and [Formula: see text] is a large sparse random matrix, accounting for the interactions between species. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Bizeul and Najim (2021).

Coexistence in diverse communities with higher-order interactions

A central assumption in most ecological models is that the interactions in a community operate only between pairs of species. However, two species may interactively affect the growth of a focal species. Although interactions among three or more species, called higher-order interactions, have the potential to modify our theoretical understanding of coexistence, ecologists lack clear expectations for how these interactions shape community structure. Here we analytically predict and numerically confirm how the variability and strength of higher-order interactions affect species coexistence. We found that as higher-order interaction strengths became more variable across species, fewer species could coexist, echoing the behavior of pairwise models. If interspecific higher-order interactions became too harmful relative to self-regulation, coexistence in diverse communities was destabilized, but coexistence was also lost when these interactions were too weak and mutualistic higher-order effects became prevalent. This behavior depended on the functional form of the interactions as the destabilizing effects of the mutualistic higher-order interactions were ameliorated when their strength saturated with species' densities. Last, we showed that more species-rich communities structured by higher-order interactions lose species more readily than their species-poor counterparts, generalizing classic results for community stability. Our work provides needed theoretical expectations for how higher-order interactions impact species coexistence in diverse communities.

Local and collective transitions in sparsely-interacting ecological communities

Interactions in natural communities can be highly heterogeneous, with any given species interacting appreciably with only some of the others, a situation commonly represented by sparse interaction networks. We study the consequences of sparse competitive interactions, in a theoretical model of a community assembled from a species pool. We find that communities can be in a number of different regimes, depending on the interaction strength. When interactions are strong, the network of coexisting species breaks up into small subgraphs, while for weaker interactions these graphs are larger and more complex, eventually encompassing all species. This process is driven by the emergence of new allowed subgraphs as interaction strength decreases, leading to sharp changes in diversity and other community properties, and at weaker interactions to two distinct collective transitions: a percolation transition, and a transition between having a unique equilibrium and having multiple alternative equilibria. Understanding community structure is thus made up of two parts: first, finding which subgraphs are allowed at a given interaction strength, and secondly, a discrete problem of matching these structures over the entire community. In a shift from the focus of many previous theories, these different regimes can be traversed by modifying the interaction strength alone, without need for heterogeneity in either interaction strengths or the number of competitors per species.

Network analysis suggests changes in food web stability produced by bottom trawl fishery in Patagonia

Demersal fisheries are one of the top anthropic stressors in marine environments. In the long term, some species are more vulnerable to fishery impacts than others, which can lead to permanent changes on the food web. The trophic relationships between predator and prey constitute the food web and it represents a network of the energy channels in an ecosystem. In turn, the network structure influences ecosystem diversity and stability. The first aim of this study was to describe for the first time the food web of the San Jorge Gulf (Patagonia Argentina) with high resolution, i.e. to the species level when information is available. The San Jorge Gulf was subject to intense fisheries thus our second aim is to analyse the food web structure with and without fishery to evaluate if the bottom-trawl industrial fishery altered the network structure and stability. We used several network metrics like: mean trophic level, omnivory, modularity and quasi-sign stability. We included these metrics because they are related to stability and can be evaluated using predator diets that can weight the links between predators and prey. The network presented 165 species organized in almost five trophic levels. The inclusion of a fishery node adds 69 new trophic links. All weighted and unweighted metrics showed differences between the two networks, reflecting a decrease in stability when fishery was included in the system. Thus, our results suggested a probable change of state of the system. The observed changes in species abundances since the fishery was established, could represent the state change predicted by network analysis. Our results suggests that changes in the stability of food webs can be used to evaluate the impacts of human activity on ecosystems.

Sustainability of the rice-crayfish co-culture aquaculture model: microbiome profiles based on multi-kingdom analyses

While the rice-crayfish culture (RCFP) model, an important aquaculture model in Asia, is generally considered a sustainable model, its sustainability in terms of microbial community profiles has not been evaluated. In this study, multi-kingdom analyses of microbiome profiles (i.e., bacteria, archaea, viruses, and eukaryotes) were performed using environmental (i.e., water and sediment) and animal gut (i.e., crayfish and crab gut) microbial samples from the RCFP and other aquaculture models, including the crab-crayfish co-culture, crayfish culture, and crab culture models, to evaluate the sustainability of the RCFP systematically. Results showed that RCFP samples are enriched with a distinct set of microbes, including Shewanella, Ferroplasma, Leishmania, and Siphoviridae, when compared with other aquaculture models. Additionally, most microbes in the RCFP samples, especially microbes from different kingdoms, were densely and positively connected, which indicates their robustness against environmental stress. Whereas microbes in different aquaculture models demonstrated moderate levels of horizontal gene transfer (HGT) across kingdoms, the RCFP showed relatively lower frequencies of HGT events, especially those involving antibiotic resistance genes. Finally, environmental factors, including pH, oxidation-reduction potential, temperature, and total nitrogen, contributed profoundly to shaping the microbial communities in these aquaculture models. Interestingly, compared with other models, the microbial communities of the RCFP model were less influenced by these environmental factors, which suggests that microbes in the latter have stronger ability to resist environmental stress. The findings collectively reflect the unique multi-kingdom microbial patterns of the RCFP model and suggest that this model is a sustainable model from the perspective of microbiome profiles.

Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.

High-order correlations in species interactions lead to complex diversity-stability relationships for ecosystems

How ecosystems maintain stability is an active area of research. Inspired by applications of random matrix theory in nuclear physics, May showed decades ago that in an ecosystem model with many randomly interacting species, increasing species diversity decreases the stability of the ecosystem. There have since been many additions to May's efforts, one being an improved understanding the effect of mutualistic, competitive, or predator-prey-like correlations between pairs of species. Here we extend a random matrix technique developed in the context of spin-glass theory to study the effect of high-order correlations among species interactions. The resulting analytically solvable models include next-to-nearest-neighbor correlations in the species interaction network, such as the enemy of my enemy is my friend, as well as higher-order correlations. We find qualitative differences from May and others' models, including nonmonotonic diversity-stability relationships. Furthermore, inclusion of particular next-to-nearest-neighbor correlations in predator-prey as opposed to mutualist-competitive networks causes the former to transition to being more stable at higher species diversity. We discuss potential applicability of our results to microbiota engineering and to the ecology of interpredator interactions, such as cub predation between lions and hyenas as well as companionship between humans and dogs.

Spatial heterogeneity enhance robustness of large multi-species ecosystems

Understanding ecosystem stability and functioning is a long-standing goal in theoretical ecology, with one of the main tools being dynamical modelling of species abundances. With the help of spatially unresolved (well-mixed) population models and equilibrium dynamics, limits to stability and regions of various ecosystem robustness have been extensively mapped in terms of diversity (number of species), types of interactions, interaction strengths, varying interaction networks (for example plant-pollinator, food-web) and varying structures of these networks. Although many insights have been gained, the impact of spatial extension is not included in this body of knowledge. Recent studies of spatially explicit modelling on the other hand have shown that stability limits can be crossed and diversity increased for systems with spatial heterogeneity in species interactions and/or chaotic dynamics. Here we show that such crossing and diversity increase can appear under less strict conditions. We find that the mere possibility of varying species abundances at different spatial locations make possible the preservation or increase in diversity across previous boundaries thought to mark catastrophic transitions. In addition, we introduce and make explicit a multitude of different dynamics a spatially extended complex system can use to stabilise. This expanded stabilising repertoire of dynamics is largest at intermediate levels of dispersal. Thus we find that spatially extended systems with intermediate dispersal are more robust, in general have higher diversity and can stabilise beyond previous stability boundaries, in contrast to well-mixed systems.

One of the major challenges facing humanity is the fragmentation of wildlife habitats and decline in biodiversity due to human land-use practices and need for resources. We need to find ways to combine human prosperity with biodiversity conservation. To achieve this a solid understanding of ecosystem stability and functioning is paramount. One way to gain such insight is to find limits when we expect species to go extinct or ecosystems to collapse by simulations of interacting species populations. Many such stability limits have been found theoretically the last decades, but for simplification of modelling, studies often exclude that ecosystems are spread out in space. Here, we explicitly include space and thus allow for dispersal and spatial heterogeneity (local differences) in species abundances. We find that for an ecosystem with the possibility of local spatial heterogeneity, the repertoire of the system’s dynamical behaviour increases dramatically. This increase in possibilities increases system robustness, enables limits previously marking extinction or collapse to be crossed without any remarkable change in global species abundances, and increases biodiversity. Thus we elucidate an additional mechanism pointing to spatial heterogeneity as crucial for ecosystem stability. We find intermediate dispersal as the most favourable for robustness and diversity of ecosystems since they display the largest repertoire of dynamical behaviour.

Merging dynamical and structural indicators to measure resilience in multispecies systems

Resilience is broadly understood as the ability of an ecological system to resist and recover from perturbations acting on species abundances and on the system's structure. However, one of the main problems in assessing resilience is to understand the extent to which measures of recovery and resistance provide complementary information about a system. While recovery from abundance perturbations has a strong tradition under the analysis of dynamical stability, it is unclear whether this same formalism can be used to measure resistance to structural perturbations (e.g. perturbations to model parameters). Here, we provide a framework grounded on dynamical and structural stability in Lotka-Volterra systems to link recovery from small perturbations on species abundances (i.e. dynamical indicators) with resistance to parameter perturbations of any magnitude (i.e. structural indicators). We use theoretical and experimental multispecies systems to show that the faster the recovery from abundance perturbations, the higher the resistance to parameter perturbations. We first use theoretical systems to show that the return rate along the slowest direction after a small random abundance perturbation (what we call full recovery) is negatively correlated with the largest random parameter perturbation that a system can withstand before losing any species (what we call full resistance). We also show that the return rate along the second fastest direction after a small random abundance perturbation (what we call partial recovery) is negatively correlated with the largest random parameter perturbation that a system can withstand before at most one species survives (what we call partial resistance). Then, we use a dataset of experimental microbial systems to confirm our theoretical expectations and to demonstrate that full and partial components of resilience are complementary. Our findings reveal that we can obtain the same level of information about resilience by measuring either a dynamical (i.e. recovery) or a structural (i.e. resistance) indicator. Irrespective of the chosen indicator (dynamical or structural), our results show that we can obtain additional information by separating the indicator into its full and partial components. We believe these results can motivate new theoretical approaches and empirical analyses to increase our understanding about risk in ecological systems.

Behaviour moderates the impacts of food-web structure on species coexistence

How species coexistence (mathematical 'feasibility') in food webs emerges from species' trophic interactions remains a long-standing open question. Here we investigate how structure (network topology and body-size structure) and behaviour (foraging strategy and spatial dimensionality of interactions) interactively affect feasibility in food webs. Metabolically-constrained modelling of food-web dynamics based on whole-organism consumption revealed that feasibility is promoted in systems dominated by large-eat-small foraging (consumers eating smaller resources) whenever (1) many top consumers are present, (2) grazing or sit-and-wait foraging strategies are common, and (3) species engage in two-dimensional interactions. Congruently, the first two conditions were associated with dominance of large-eat-small foraging in 74 well-resolved (primarily aquatic) real-world food webs. Our findings provide a new, mechanistic understanding of how behavioural properties can modulate the effects of structural properties on species coexistence in food webs, and suggest that 'being feasible' constrains the spectra of behavioural and structural properties seen in natural food webs.

Component response rate variation underlies the stability of highly complex finite systems

The stability of a complex system generally decreases with increasing system size and interconnectivity, a counterintuitive result of widespread importance across the physical, life, and social sciences. Despite recent interest in the relationship between system properties and stability, the effect of variation in response rate across system components remains unconsidered. Here I vary the component response rates (γ) of randomly generated complex systems. I use numerical simulations to show that when component response rates vary, the potential for system stability increases. These results are robust to common network structures, including small-world and scale-free networks, and cascade food webs. Variation in γ is especially important for stability in highly complex systems, in which the probability of stability would otherwise be negligible. At such extremes of simulated system complexity, the largest stable complex systems would be unstable if not for variation in γ. My results therefore reveal a previously unconsidered aspect of system stability that is likely to be pervasive across all realistic complex systems.

Predicting collapse of complex ecological systems: quantifying the stability-complexity continuum

Dynamical shifts between the extremes of stability and collapse are hallmarks of ecological systems. These shifts are limited by and change with biodiversity, complexity, and the topology and hierarchy of interactions. Most ecological research has focused on identifying conditions for a system to shift from stability to any degree of instability-species abundances do not return to exact same values after perturbation. Real ecosystems likely have a continuum of shifting between stability and collapse that depends on the specifics of how the interactions are structured, as well as the type and degree of disturbance due to environmental change. Here we map boundaries for the extremes of strict stability and collapse. In between these boundaries, we find an intermediate regime that consists of single-species extinctions, which we call the extinction continuum. We also develop a metric that locates the position of the system within the extinction continuum-thus quantifying proximity to stability or collapse-in terms of ecologically measurable quantities such as growth rates and interaction strengths. Furthermore, we provide analytical and numerical techniques for estimating our new metric. We show that our metric does an excellent job of capturing the system's behaviour in comparison with other existing methods-such as May's stability criteria or critical slowdown. Our metric should thus enable deeper insights about how to classify real systems in terms of their overall dynamics and their limits of stability and collapse.

An Approach to Study Species Persistence in Unconstrained Random Networks

The connection between structure and stability of ecological networks has been widely studied in the last fifty years. A challenge that scientists continue to face is that in-depth mathematical model analysis is often difficult, unless the considered systems are specifically constrained. This makes it challenging to generalize results. Therefore, methods are needed that relax the required restrictions. Here, we introduce a novel heuristic approach that provides persistence estimates for random systems without limiting the admissible parameter range and system behaviour. We apply our approach to study persistence of species in random generalized Lotka-Volterra systems and present simulation results, which confirm the accuracy of our predictions. Our results suggest that persistence is mainly driven by the linkage density, whereby additional links can both favour and hinder persistence. In particular, we observed "persistence bistability", a rarely studied feature of random networks, leading to a dependency of persistence on initial species densities. Networks with this property exhibit tipping points, in which species loss can lead to a cascade of extinctions. The methods developed in this paper may facilitate the study of more general models and thereby provide a step forward towards a unifying framework of network architecture and stability.

Effect of population abundances on the stability of large random ecosystems

Random matrix theory successfully connects the structure of interactions of large ecological communities to their ability to respond to perturbations. One of the most debated aspects of this approach is that so far studies have neglected the role of population abundances on stability. While species abundances are well studied and empirically accessible, studies on stability have so far failed to incorporate this information. Here we tackle this question by explicitly including population abundances in a random matrix framework. We derive an analytical formula that describes the spectrum of a large community matrix for arbitrary feasible species abundance distributions. The emerging picture is remarkably simple: while population abundances affect the rate to return to equilibrium after a perturbation, the stability of large ecosystems is uniquely determined by the interaction matrix. We confirm this result by showing that the likelihood of having a feasible and unstable solution in the Lotka-Volterra system of equations decreases exponentially with the number of species for stable interaction matrices.