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Bacterial communities are pivotal to maintaining ecological function and preserving the rich tapestry of biological diversity. The rapid development of environmental sequencing technologies, such as metagenomics, has revolutionized our capacity to probe such diversity. However, despite these advances, a theoretical understanding connecting empirical data with ecosystem modelling, in particular in the framework of disordered systems akin to spin glasses, is still in its infancy. Here, we present a comprehensive framework using theories of disordered systems to decode microbiome data, which offers insight into the ecological forces that shape macroecological states. By employing the quenched disordered generalized Lotka-Volterra model, we analyze species abundance data in healthy and diseased human gut microbiomes. Results reveal the emergence of two distinct patterns of species-interaction networks, elucidating the pathways through which dysbiosis may drive microbiome instability. Interaction patterns thus provide a window into the systemic shifts accompanying the transition from health to disease, offering a new perspective on the dynamics of the microbial community. Our findings suggest the potential of disordered systems theory to characterize microbiomes by capturing the essence of ecological interactions and their consequences on stability and functioning, leveraging our understanding of the linkages of dysbiosis and microbial dynamics., Comment: 10 pages, 4 figures

We investigate a disordered multi-dimensional linear system in which the interaction parameters vary stochastically in time with defined temporal correlations. We refer to this type of disorder as "annealed", in contrast to quenched disorder in which couplings are fixed in time. We extend Dynamical Mean-Field Theory to accommodate annealed disorder and employ it to find the exact solution of the linear model in the limit of a large number of degrees of freedom. Our analysis yields analytical results for the non-stationary auto-correlation, the stationary variance, the power spectral density, and the phase diagram of the model. Interestingly, some unexpected features emerge upon changing the correlation time of the interactions. The stationary variance of the system and the critical variance of the disorder are generally found to be a non-monotonic function of the correlation time of the interactions. We also find that in some cases a re-entrant phase transition takes place when this correlation time is varied., Comment: 20 pages, 7 figures

The management of common-pool resources is a complex challenge due to the risk of overexploitation and the tragedy of the commons. A novel framework has been introduced to address this issue, focusing on the coevolutionary relationship between human behavior and common-pool resources within a human-environment system. However, the impact of the Allee effect on the coevolution and its resource sustainability is still unexplored. The Allee effect, a biological phenomenon characterized by a correlation between resource availability and growth rate, is a fundamental attribute of numerous natural resources. In this paper, we introduce two coevolutionary models of resource and strategy under replicator dynamics and knowledge feedback by applying the Allee effect to the common-pool resources within human-environment system. These models encapsulate various facets of resource dynamics and the players' behavior, such as resource growth function, the extraction rates, and the strategy update rules. We find that the Allee effect can induce bi-stability and critical transition, leading to either sustainable or unsustainable outcomes depending on the initial condition and parameter configuration. We demonstrate that knowledge feedback enhances the resilience and sustainability of the coevolving system, and these results advances the understanding of human-environment system and management of common-pool resources.

Metapopulation models have been instrumental in demonstrating the ecological impact of landscape structure on the survival of a focal species in complex environments. However, extensions to multiple species with arbitrary dispersal networks often rely on phenomenological assumptions limiting their scope. Here, we develop a multilayer network model of competitive dispersing metacommunities to investigate how spatially structured environments impact species coexistence and ecosystem stability. We show that homogeneous environments always lead to monodominance unless all species' fitness parameters are in an exact trade-off. However, this precise fine-tuning does not guarantee coexistence in generic heterogeneous environments. By introducing general spatial disorder in the model, we solve it exactly in the mean-field limit, finding that stable coexistence becomes possible in the presence of strong disorder. Crucially, coexistence is supported by the spontaneous localization of species through the emergence of ecological niches. Our results remain qualitatively valid in arbitrary dispersal networks, where topological features can improve species coexistence. Finally, we employ our model to study how correlated disorder promotes spatial ecological patterns in realistic terrestrial and riverine landscapes. Our work provides a novel framework to understand how landscape structure enables coexistence in metacommunities by acting as the substrate for ecological interactions., Comment: 11 pages, 4 figures

Games with environmental feedback have become a crucial area of study across various scientific domains, modelling the dynamic interplay between human decisions and environmental changes, and highlighting the consequences of our choices on natural resources and biodiversity. In this work, we propose a co-evolutionary model for human-environment systems that incorporates the effects of knowledge feedback and social interaction on the sustainability of common pool resources. The model represents consumers as agents who adjust their resource extraction based on the resource's state. These agents are connected through social networks, where links symbolize either affinity or aversion among them. The interplay between social dynamics and resource dynamics is explored, with the system's evolution analyzed across various network topologies and initial conditions. We find that knowledge feedback can independently sustain common pool resources. However, the impact of social interactions on sustainability is dual-faceted: it can either support or impede sustainability, influenced by the network's connectivity and heterogeneity. A notable finding is the identification of a critical network mean degree, beyond which a depletion/repletion transition parallels an absorbing/active state transition in social dynamics, i.e., individual agents and their connections are/are not prone to being frozen in their social states. Furthermore, the study examines the evolution of the social network, revealing the emergence of two polarized groups where agents within each community have the same affinity. Comparative analyses using Monte-Carlo simulations and rate equations are employed, along with analytical arguments, to reinforce the study's findings. The model successfully captures how information spread and social dynamics may impact the sustanebility of common pool resource.

Laboratory experiments with bacterial colonies, under well-controlled conditions often lead to evolutionary diversification, where at least two ecotypes emerge from an initially monomorphic population. Empirical evidence suggests that such "evolutionary branching" occurs stochastically, even under fixed and stable conditions. This stochastic nature is characterized by: (i) occurrence in a significant fraction, but not all, of experimental settings, (ii) emergence at widely varying times, and (iii) variable relative abundances of the resulting subpopulations across experiments. Theoretical approaches to understanding evolutionary branching under these conditions have been previously developed within the (deterministic) framework of "adaptive dynamics." Here, we advance the understanding of the stochastic nature of evolutionary outcomes by introducing the concept of "stochastic trade-offs" as opposed to "hard" ones. The key idea is that the stochasticity of mutations occurs in a high-dimensional trait space and this translates into variability that is constrained to a flexible tradeoff curve. By incorporating this additional source of stochasticity, we are able to account for the observed empirical variability and make predictions regarding the likelihood of evolutionary branching under different conditions. This approach effectively bridges the gap between theoretical predictions and experimental observations, providing insights into when and how evolutionary branching is more likely to occur in laboratory experiments.

In this work, we explore the dynamics of species abundances within ecological communities using the Generalized Lotka-Volterra (GLV) model. At variance with previous approaches, we present an analysis of stochastic GLV dynamics with temporal fluctuations in interaction strengths between species. We develop a dynamical mean field theory (DMFT) tailored for scenarios with annealed colored noise and simple functional responses. We show that time-dependent interactions can be effectively modeled as environmental noise in the DMFT and we obtain analytical predictions for the species abundance distribution that well matches empirical observations. Our results suggest that environmental noise favors species coexistence and allows to overcome the complexity-stability paradox, especially in comparison to dynamics with quenched disorder. This study offers new insights not only into the modeling of large ecosystem dynamics, but also proposes novel methodologies for examining ecological systems., Comment: 4 Figures

Biological and living systems process information across spatiotemporal scales, exhibiting the hallmark ability to constantly modulate their behavior to ever-changing and complex environments. In the presence of repeated stimuli, a distinctive response is the progressive reduction of the activity at both sensory and molecular levels, known as habituation. Here, we solve a minimal microscopic model devoid of biological details to show that habituation is driven by negative feedback provided by a slow storage mechanism. Crucially, an intermediate level of habituation is associated with a steep increase in the information that the system collects on the external input over time. We find that the region characterized both by maximal information gain and by the onset of habituation can be retrieved if the system tunes its parameters to minimize dissipation and maximize information at the same time. We test our dynamical predictions against experimentally recorded neural responses in a zebrafish larva subjected to repeated looming stimulation. Our work makes a fundamental step towards uncovering the core mechanisms that shape habituation in biological systems, elucidating its information-theoretic and functional role.

Molecular ecology uses molecular genetic data to answer traditional ecological questions in biogeography and biodiversity among others. Several ecological principles, such as the niche hypothesis and the competitive exclusions, are based on the fact that species compete for resources. More in general, it is now recognized that species interactions play a crucial role in determining the coexistence and abundance of species. However, experimentally controllable platforms, which allow to study and measure competitions among species, are rare and difficult to implement. In this work, we suggest to exploit a Molecular Dynamics coarse-grained model to study interactions among single strands of DNA, representing individuals of different species, which compete for binding to other oligomers considered as resources. In particular, the well-established knowledge of DNA-DNA interactions at the nanoscale allows us to test the hypothesis that the maximum consecutive overlap between pairs of oligomers measure the species competitive advantages. However, we suggest that more complex structure also plays a role in the ability of the species to successfully bind to the target resource oligomer. We complement the simulations with experiments on populations of DNA strands which qualitatively confirm our hypotheses. These tools constitute a promising starting point for further developments concerning the study of controlled, DNA-based, artificial ecosystems.

Understanding the conditions of feasibility and stability in ecological systems is a major challenge in theoretical ecology. The seminal work of May in 1972 and recent developments based on the theory of random matrices have shown the existence of emergent universal patterns of both stability and feasibility in ecological dynamics. However, only a few studies have investigated the role of delay coupled with population dynamics in the emergence of feasible and stable states. In this work, we study the effects of delay on Generalized Loka-Volterra population dynamics of several interacting species in closed ecological environments. First, we investigate the relation between feasibility and stability of the modeled ecological community in the absence of delay and find a simple analytical relation when intra-species interactions are dominant. We then show how, by increasing the time delay, there is a transition in the stability phases of the population dynamics: from an equilibrium state to a stable non-point attractor phase. We calculate analytically the critical delay of that transition and show that it is in excellent agreement with numerical simulations. Finally, we introduce a measure of stability that holds for out of equilibrium dynamics and we show that in the oscillatory regime induced by the delay stability increases for increasing ecosystem diversity.

Between 2019 and 2020, during the country's hottest and driest year on record, Australia experienced a dramatic bushfire season, with catastrophic ecological and environmental consequences. Several studies highlighted how such abrupt changes in fire regimes may have been in large part a consequence of climate change and other anthropogenic transformations. Here, we analyze the monthly evolution of the burned area in Australia from 2000 to 2020, obtained via satellite imaging through the MODIS platform. We find that the 2019-2020 peak is associated with signatures typically found near critical points. We introduce a modeling framework based on forest-fire models to study the properties of these emergent fire outbreaks, showing that the behavior observed during the 2019-2020 fire season matches the one of a percolation transition, where system-size outbreaks appear. Our model also highlights the existence of an absorbing phase transition that might be eventually crossed, after which the vegetation cannot recover.

The sustainable use of common-pool resources (CPRs) is a major environmental governance challenge because of their possible over-exploitation. Research in this field has overlooked the feedback between user decisions and resource dynamics. Here we develop an online game to perform a set of experiments in which users of the same CPR decide on their individual harvesting rates, which in turn depend on the resource dynamics. We show that, if users share common goals, a high level of self-organized cooperation emerges, leading to long-term resource sustainability. Otherwise, selfish/individualistic behaviors lead to resource depletion ("Tragedy of the Commons"). To explain these results, we develop an analytical model of coupled resource-decision dynamics based on optimal control theory and show how this framework reproduces the empirical results.

The critical brain hypothesis has emerged as an attractive framework to understand neuronal activity, but it is still widely debated. In this work, we analyze data from a multi-electrodes array in the rat's cortex and we find that power-law neuronal avalanches satisfying the crackling-noise relation coexist with spatial correlations that display typical features of critical systems. In order to shed a light on the underlying mechanisms at the origin of these signatures of criticality, we introduce a paradigmatic framework with a common stochastic modulation and pairwise linear interactions inferred from our data. We show that in such models power-law avalanches that satisfy the crackling noise relation emerge as a consequence of the extrinsic modulation, whereas scale-free correlations are solely determined by internal interactions. Moreover, this disentangling is fully captured by the mutual information in the system. Finally, we show that analogous power-law avalanches are found in more realistic models of neural activity as well, suggesting that extrinsic modulation might be a broad mechanism for their generation.

Local coexistence of species in large ecosystems is traditionally explained within the broad framework of niche theory. However, its rationale hardly justifies rich biodiversity observed in nearly homogeneous environments. Here we consider a consumer-resource model in which a coarse-graining procedure accounts for a variety of ecological mechanisms and leads to effective spatial effects which favour species coexistence. Herein, we provide conditions for several species to live in an environment with very few resources. In fact, the model displays two different phases depending on whether the number of surviving species is larger or smaller than the number of resources. We obtain conditions whereby a species can successfully colonize a pool of coexisting species. Finally, we analytically compute the distribution of the population sizes of coexisting species. Numerical simulations as well as empirical distributions of population sizes support our analytical findings., Comment: 5 pages + 4 figures, Major revision

We used an ecological approach based on a neutral model to study the competition for attention in an online social network. This novel approach allow us to analyze some ecological patterns that has also an insightful meaning in the context of information ecosystem. Specifically, we focus on the study of patterns related with the persistence of a meme within the network and the capacity of the system to sustain coexisting memes. Not only are we able of doing such analysis in an approximated continuum limit, but also we get exact results of the finite-size discrete system.

Human perceptual and cognitive abilities are limited resources. Today, in the age of cheap information --cheap to produce, to manipulate, to disseminate--, this cognitive bottleneck translates into hypercompetition for visibility among actors (individuals, institutions, etc). The same social communication incentive --visibility-- pushes actors to mutualistically interact with specific memes, seeking the virality of their messages. In turn, contents are driven by selective pressure, i.e. the chances to persist and reach widely are tightly subject to changes in the communication environment. In spite of all this complexity, here we show that the underlying architecture of the users-memes interaction in information ecosystems, apparently chaotic and noisy, actually evolves towards emergent patterns, reminiscent of those found in natural ecosystems. In particular we show, through the analysis of empirical, large data streams, that communication networks are structurally elastic, i.e. fluctuating from modular to nested architecture as a response to environmental perturbations (e.g. extraordinary events). We then propose an ecology-inspired modelling framework, bringing to light the precise mechanisms causing the observed dynamical reorganisation. Finally, from numerical simulations, the model predicts --and the data confirm-- that the users' struggle for visibility induces a re-equilibration of the network towards a very constrained organisation: the emergence of self-similar nested arrangements., Comment: Main text and supplementary information

Neuromorphic networks can be described in terms of coarse-grained variables, where emergent sustained behaviours spontaneously arise if stochasticity is properly taken in account. For example it has been recently found that a directed linear chain of connected patch of neurons amplifies an input signal, also tuning its characteristic frequency. Here we study a generalization of such a simple model, introducing heterogeneity and variability in the parameter space and long-range interactions, breaking, in turn, the preferential direction of information transmission of a directed chain. On one hand, enlarging the region of parameters leads to a more complex state space that we analytically characterise; moreover, we explicitly link the strength distribution of the non-local interactions with the frequency distribution of the network oscillations. On the other hand, we found that adding long-range interactions can cause the onset of novel phenomena, as coherent and synchronous oscillations among all the interacting units, which can also coexist with the amplification of the signal.

We present a systematic study to test a recently introduced phenomenological renormalization group, proposed to coarse-grain data of neural activity from their correlation matrix. The approach allows, at least in principle, to establish whether the collective behavior of the network of spiking neurons is described by a non-Gaussian critical fixed point. We test this renormalization procedure in a variety of models focusing in particular on the contact process, which displays an absorbing phase transition at $\lambda = \lambda_c$ between a silent and an active state. We find that the results of the coarse-graining do not depend on the presence of long-range interactions, but some scaling features persist in the super-critical system up to a distance of $10\%$ from $\lambda_c$. Our results provide insights on the possible subtleties that one needs to consider when applying such phenomenological approaches directly to data to infer signatures of criticality., Comment: 9 pages, 8 figures

In recent years we have witnessed an explosion of data collected for different human dynamics, from email communication to social networks activities. Extract useful information from these huge data sets represents a major challenge. In the last decades, statistical regularities has been widely observed in human activities and various models have been proposed. Here we move from modeling to inference and propose a statistical framework capable to predict global features of human activities from local knowledge. We consider four data sets of human activities: email communication, Twitter posts, Wikipedia articles and Gutenberg books. From the statistics of local activities, such as sent emails per senders, post per hashtags and word occurrences collected in a small sample of the considered dataset, we infer global features, as the number of senders, hashtags and words at the global scale. Our estimates are robust and accurate with a small relative error. Moreover, we predict how abundance of a hashtag or of a word may change through scales. Thus, observing a small portion of tweets and the popularity of a given hashtag among them, we can estimate whether it will remain popular or not in the unseen part of the network. Our approach is grounded on statistical ecology as we discover inference of unseen human activity hallmarks can be mapped into the unseen species problem in biodiversity. Our findings may have applications to different areas, from resource management in emails to collective attention monitoring in Twitter and to language learning process in word databases.

In our recent article (Tu et al., Warnings and caveats in brain controllability, arXiv:1705.08261) we provided quantitative evidence to show that there are warnings and caveats in the way Gu and collaborators (Gu et al. Controllability of structural brain networks. Nature communications 6 (2015): 8414) define brain controllability. The comment by Pasqualetti et al. (Pasqualetti et al. RE: Warnings and Caveats in Brain Controllability. NeuroImage 297 (2019), 586-588) confirms the need to go beyond the methodology and approach presented in Gu et al. original work. In fact, they recognize that the source of confusion is due to the fact that assessing controllability via numerical analysis typically leads to ill-conditioned problems, and thus often generates results that are difficult to interpret. This is indeed the first warning we discussed: our work was not meant to prove that brain networks are not controllable from one node, rather we wished to highlight that the one node controllability framework and all consequent results were not properly justified based on the methodology presented in Gu et al. We used in our work the same method of Gu et al. not because we believe it is the best methodology, but because we extensively investigated it with the aim of replicating, testing and extending their results. And the warning and caveats we have proposed are the results of this investigation., Comment: 3 pages, 1 figure

We analyze about two hundred naturally occurring networks with distinct dynamical origins to formally test whether the commonly assumed hypothesis of an underlying scale-free structure is generally viable. This has recently been questioned on the basis of statistical testing of the validity of power law distributions of network degrees by contrasting real data. Specifically, we analyze by finite-size scaling analysis the datasets of real networks to check whether purported departures from the power law behavior are due to the finiteness of the sample size. In this case, power laws would be recovered in the case of progressively larger cutoffs induced by the size of the sample. We find that a large number of the networks studied follow a finite size scaling hypothesis without any self-tuning. This is the case of biological protein interaction networks, technological computer and hyperlink networks, and informational networks in general. Marked deviations appear in other cases, especially infrastructure and transportation but also social networks. We conclude that underlying scale invariance properties of many naturally occurring networks are extant features often clouded by finite-size effects due to the nature of the sample data.

The evolutionary and ecological processes behind the origin of species are among the most fundamental problems in biology. In fact, many theoretical hypothesis on different type of speciation have been proposed. In particular, models of sympatric speciation leading to the formation of new species without geographical isolation, are based on the niche hypothesis: the diversification of the population is induced by the competition for a limited set of the available resources. On the other hand, neutral models of evolution have shown that stochastic forces are sufficient to generate coexistence of different species. In this work, we bring this dichotomy to the context of species formation, and we study how neutral and niche forces contribute to sympatric speciation in a model ecosystem. In particular, we study the evolution of a population of individuals with asexual reproduction whose inherited characters or phenotypes are specified by both niche-based and neutral traits. We analyse the stationary state of the dynamics, and study the distribution of individuals in the whole space of possible phenotypes. We show, both by numerical simulations and analytics, that there is a non-trivial coupling between neutral and niche forces induced by stochastic effects in the evolution of the population that allows the formation of clusters (i.e., species) in the phenotypic space. Our framework can be generalised also to sexual reproduction or other type of population dynamics., Comment: 8 pages, 5 figures

Thanks to the availability of large scale digital datasets and massive amounts of computational power, deep learning algorithms can learn representations of data by exploiting multiple levels of abstraction. These machine learning methods have greatly improved the state-of-the-art in many challenging cognitive tasks, such as visual object recognition, speech processing, natural language understanding and automatic translation. In particular, one class of deep learning models, known as deep belief networks, can discover intricate statistical structure in large data sets in a completely unsupervised fashion, by learning a generative model of the data using Hebbian-like learning mechanisms. Although these self-organizing systems can be conveniently formalized within the framework of statistical mechanics, their internal functioning remains opaque, because their emergent dynamics cannot be solved analytically. In this article we propose to study deep belief networks using techniques commonly employed in the study of complex networks, in order to gain some insights into the structural and functional properties of the computational graph resulting from the learning process., Comment: 20 pages, 9 figures

In order to measure, predict, and prevent social segregation, it is necessary to understand the factors that cause it. While in most available descriptions space plays an essential role, one outstanding question is whether and how this phenomenon is possible in a well-mixed social network. We define and solve a simple model of segregation on networks based on discrete convictions. In our model, space does not play a role, and individuals never change their conviction, but they may choose to connect socially to other individuals based on two criteria: sharing the same conviction, and individual popularity (regardless of conviction). The trade-off between these two moves defines a parameter, analogous to the "tolerance" parameter in classical models of spatial segregation. We show numerically and analytically that this parameter determines a true phase transition (somewhat reminiscent of phase separation in a binary mixture) between a well-mixed and a segregated state. Additionally, minority convictions segregate faster and inter-specific aversion alone may lead to a segregation threshold with similar properties. Together, our results highlight the general principle that a segregation transition is possible in absence of spatial degrees of freedom, provided that conviction-based rewiring occurs on the same time scale of popularity rewirings., Comment: 11 pages, 8 figures

The year 2017 saw the rise and fall of the crypto-currency market, followed by high variability in the price of all crypto-currencies. In this work, we study the abrupt transition in crypto-currency residuals, which is associated with the critical transition (the phenomenon of critical slowing down) or the stochastic transition phenomena. We find that, regardless of the specific crypto-currency or rolling window size, the autocorrelation always fluctuates around a high value, while the standard deviation increases monotonically. Therefore, while the autocorrelation does not display signals of critical slowing down, the standard deviation can be used to anticipate critical or stochastic transitions. In particular, we have detected two sudden jumps in the standard deviation, in the second quarter of 2017 and at the beginning of 2018, which could have served as early warning signals of two majors price collapses that have happened in the following periods. We finally propose a mean-field phenomenological model for the price of crypto-currency to show how the use of the standard deviation of the residuals is a better leading indicator of the collapse in price than the time series' autocorrelation. Our findings represent a first step towards a better diagnostic of the risk of critical transition in the price and/or volume of crypto-currencies., Comment: 14 pages, 5 figures, 1 table

Empirical observations show that ecological communities can have a huge number of coexisting species, also with few or limited number of resources. These ecosystems are characterized by multiple type of interactions, in particular displaying cooperative behaviors. However, standard modeling of population dynamics based on Lotka-Volterra type of equations predicts that ecosystem stability should decrease as the number of species in the community increases and that cooperative systems are less stable than communities with only competitive and/or exploitative interactions. Here we propose a stochastic model of population dynamics, which includes exploitative interactions as well as cooperative interactions induced by cross-feeding. The model is exactly solved and we obtain results for relevant macro-ecological patterns, such as species abundance distributions and correlation functions. In the large system size limit, any number of species can coexist for a very general class of interaction networks and stability increases as the number of species grows. For pure mutualistic/commensalistic interactions we determine the topological properties of the network that guarantee species coexistence. We also show that the stationary state is globally stable and that inferring species interactions through species abundance correlation analysis may be misleading. Our theoretical approach thus show that appropriate models of cooperation naturally leads to a solution of the long-standing question about complexity-stability paradox and on how highly biodiverse communities can coexist., Comment: Incorrectdly posted twice. Current version arXiv:1708.03154

Understanding the relationship between large-scale structural and functional brain networks remains a crucial issue in modern neuroscience. Recently, there has been growing interest in investigating the role of homeostatic plasticity mechanisms, across different spatiotemporal scales, in regulating network activity and brain functioning against a wide range of environmental conditions and brain states (e.g., during learning, development, ageing, neurological diseases). In the present study, we investigate how the inclusion of homeostatic plasticity in a stochastic whole-brain model, implemented as a normalization of the incoming node's excitatory input, affects the macroscopic activity during rest and the formation of functional networks. Importantly, we address the structure-function relationship both at the group and individual-based levels. In this work, we show that normalization of the node's excitatory input improves the correspondence between simulated neural patterns of the model and various brain functional data. Indeed, we find that the best match is achieved when the model control parameter is in its critical value and that normalization minimizes both the variability of the critical points and neuronal activity patterns among subjects. Therefore, our results suggest that the inclusion of homeostatic principles lead to more realistic brain activity consistent with the hallmarks of criticality. Our theoretical framework open new perspectives in personalized brain modeling with potential applications to investigate the deviation from criticality due to structural lesions (e.g. stroke) or brain disorders., Comment: Accepted for publication in Scientific Reports

Empirical observations show that ecological communities can have a huge number of coexisting species, also with few or limited number of resources. These ecosystems are characterized by multiple type of interactions, in particular displaying cooperative behaviors. However, standard modeling of population dynamics based on Lotka-Volterra type of equations predicts that ecosystem stability should decrease as the number of species in the community increases and that cooperative systems are less stable than communities with only competitive and/or exploitative interactions. Here we propose a stochastic model of population dynamics, which includes exploitative interactions as well as cooperative interactions induced by cross-feeding. The model is exactly solved and we obtain results for relevant macro-ecological patterns, such as species abundance distributions and correlation functions. In the large system size limit, any number of species can coexist for a very general class of interaction networks and stability increases as the number of species grows. For pure mutualistic/commensalistic interactions we determine the topological properties of the network that guarantee species coexistence. We also show that the stationary state is globally stable and that inferring species interactions through species abundance correlation analysis may be misleading. Our theoretical approach thus show that appropriate models of cooperation naturally leads to a solution of the long-standing question about complexity-stability paradox and on how highly biodiverse communities can coexist., Comment: 25 pages, 10 figures

In this work we challenge the main conclusions of Gu et al work (Controllability of structural brain networks. Nature communications 6, 8414, doi:10.1038/ncomms9414, 2015) on brain controllability. Using the same methods and analyses on four datasets we find that the minimum set of nodes to control brain networks is always larger than one. We also find that the relationships between the average/modal controllability and weighted degrees also hold for randomized data and the there are not specific roles played by Resting State Networks in controlling the brain. In conclusion, we show that there is no evidence that topology plays specific and unique roles in the controllability of brain networks. Accordingly, Gu et al. interpretation of their results, in particular in terms of translational applications (e.g. using single node controllability properties to define target region(s) for neurostimulation) should be revisited. Though theoretically intriguing, our understanding of the relationship between controllability and structural brain network remains elusive., Comment: 9 pages, 1 Figure, 1 Table

The increasing volume of ecologically and biologically relevant data has revealed a wide collection of emergent patterns in living systems. Analyzing different datasets, ranging from metabolic gene-regulatory to species interaction networks, we find that these networks are sparse, i.e. the percentage of the active interactions scales inversely proportional to the system size. This puzzling characteristic has been neither yet considered nor explained. Herein, we introduce the new concept of explorability, a measure of the ability of the system to adapt to newly intervening changes. We show that sparsity is an emergent property resulting from a variational principle aiming at the optimization of both explorability and dynamical robustness, the capacity of the system to remain stable after perturbations of the underlying dynamics. Networks with higher connectivities lead to an incremental difficulty to find better values for both the explorability and dynamical robustness, associated with the fine-tuning of the newly added interactions. A relevant characteristic of our solution is its scale invariance, that is, it remains optimal when several communities are assembled togheter. Connectivity is also a key ingredient determining ecosystem stability and our proposed solution contributes to solving May's celebrated complexity-stability paradox., Comment: Supplementary Information upon request

Environmental fluctuations have important consequences in the organization of ecological communities, and understanding how such a variability influences the biodiversity of an ecosystem is a major question in ecology. In this paper, we analyze the case of two species competing for the resources within the framework of the neutral theory in the presence of environmental noise, devoting special attention on how such a variability modulates species fitness. The environment is dichotomous and stochastically alternates between periods favoring one of the species while disfavoring the other one, preserving neutrality on the long term. We study two different scenarios: in the first one species fitness varies linearly with the environment, and in the second one the effective fitness is re-scaled by the total fitness of the individuals competing for the same resource. We find that, in the former case environmental fluctuations always reduce the time of species coexistence, whereas such a time can be enhanced or reduced in the latter case, depending on the correlation time of the environment. This phenomenon can be understood as a direct consequence of Chesson's storage effect., Comment: 11 pages, 4 figures

In this work we study the likelihood of survival of single-species in the context of hostile and disordered environments. Population dynamics in this environment, as modeled by the Fisher equation, is characterized by negative average growth rate, except in some random spatially distributed patches that may support life. In particular, we are interested in the phase diagram of the survival probability and in the critical size problem, i.e., the minimum patch size required for surviving in the long time dynamics. We propose a measure for the critical patch size as being proportional to the participation ratio (PR) of the eigenvector corresponding to the largest eigenvalue of the linearized Fisher dynamics. We obtain the (extinction-survival) phase diagram and the probability distribution function (PDF) of the critical patch sizes for two topologies, namely, the one-dimensional system and the fractal Peano basin. We show that both topologies share the same qualitative features, but the fractal topology requires higher spatial fluctuations to guarantee species survival. We perform a finite-size scaling and we obtain the associated scaling exponents. In addition, we show that the PDF of the critical patch sizes has an universal shape for the 1D case in terms of the model parameters (diffusion, growth rate, etc.). In contrast, the diffusion coefficient has a drastic effect on the PDF of the critical patch sizes of the fractal Peano basin, and it does not obey the same scaling law of the 1D case., Comment: 20 pages, 5 Figures

Recently, a theoretical framework aimed at separating the roles of dynamics and topology in multi-dimensional systems has been developed (Gao et al, \textit{Nature}, Vol 530:307 (2016)). The validity of their method is assumed to hold depending on two main hypothesis: $(i)$ The network determined by the the interaction between pairs of nodes has negligible degree correlations; $(ii)$ The node activities are uniform across nodes on both the drift and pair-wise interaction functions. Moreover, the authors consider only positive (mutualistic) interactions. Here we show the conditions proposed by Gao and collaborators are neither sufficient nor necessary to guarantee that their method works in general, and validity of their results are not independent of the model chosen within the class of dynamics they considered. Indeed we find that a new condition poses effective limitations to their framework and we provide quantitative predictions of the quality of the one dimensional collapse as a function of the properties of interaction networks and stable dynamics using results from random matrix theory. We also find that multi-dimensional reduction may work also for interaction matrix with a mixture of positive and negative signs, opening up application of the framework to food-webs, neuronal networks and social/economic interactions.

The easy access to large data sets has allowed for leveraging methodology in network physics and complexity science to disentangle patterns and processes directly from the data, leading to key insights in the behavior of systems. Here we use to country specific food production data to study binary and weighted topological properties of the bipartite country-food production matrix. This country-food production matrix can be: 1) transformed into overlap matrices which embed information regarding shared production of products among countries, and or shared countries for individual products, 2) identify subsets of countries which produce similar commodities or subsets of commodities shared by a given country allowing for visualization of correlations in large networks, and 3) used to rank country's fitness (the ability to produce a diverse array of products weighted on the type of food commodities) and food specialization (quantified on the number of countries producing that food product weighted on their fitness). Our results show that, on average, countries with high fitness producing highly specialized food commodities also produce low specialization goods, while nations with low fitness producing a small basket of diverse food products, typically produce low specialized food commodities.

The hypothesis that living systems can benefit from operating at the vicinity of critical points has gained momentum in recent years. Criticality may confer an optimal balance between exceedingly ordered and too noisy states. We here present a model, based on information theory and statistical mechanics, illustrating how and why a community of agents aimed at understanding and communicating with each other converges to a globally coherent state in which all individuals are close to an internal critical state, i.e. at the borderline between order and disorder. We study --both analytically and computationally-- the circumstances under which criticality is the best possible outcome of the dynamical process, confirming the convergence to critical points under very generic conditions. Finally, we analyze the effect of cooperation (agents try to enhance not only their fitness, but also that of other individuals) and competition (agents try to improve their own fitness and to diminish those of competitors) within our setting. The conclusion is that, while competition fosters criticality, cooperation hinders it and can lead to more ordered or more disordered consensual solutions., Comment: 20 pages, 5 figures. Supplementary Material: 8 pages

The role of species interactions in controlling the interplay between the stability of an ecosystem and its biodiversity is still not well understood. The ability of ecological communities to recover after a small perturbation of the species abundances (local asymptotic stability) has been well studied, whereas the likelihood of a community to persist when the interactions are altered (structural stability) has received much less attention. Our goal is to understand the effects of diversity, interaction strenghts and ecological network structure on the volume of parameter space leading to feasible equilibria, i.e., ones in which all populations have positive abundances. We develop a geometrical framework to study the range of conditions necessary for feasible coexistence in both mutualistic and consumer-resource systems. Using analytical and numerical methods, we show that feasibility is determined by just a handful of quantities describing the interactions, yielding a nontrivial complexity-feasibility relationship. Analyzing more than 100 empirical networks, we show that the range of coexistence conditions in mutualistic systems can be analytically predicted by means of a null model of random interactions, whereas food webs are characterized by smaller coexistence domains than those expected by chance. Finally, we characterize the geometric shape of the feasibility domain, thereby identifying the direction of perturbations that are more likely to cause extinctions. Interestingly, the structure of mutualistic interactions leads to very heterogeneous responses to perturbations, making those systems more fragile than expected by chance., Comment: 61 pages, 17 figures

The simplest theories often have much merit and many limitations, and in this vein, the value of Neutral Theory (NT) has been the subject of much debate over the past 15 years. NT was proposed at the turn of the century by Stephen Hubbell to explain pervasive patterns observed in the organization of ecosystems. Its originally tepid reception among ecologists contrasted starkly with the excitement it caused among physicists and mathematicians. Indeed, NT spawned several theoretical studies that attempted to explain empirical data and predicted trends of quantities that had not yet been studied. While there are a few reviews of NT oriented towards ecologists, our goal here is to review the quantitative results of NT and its extensions for physicists who are interested in learning what NT is, what its successes are and what important problems remain unresolved. Furthermore, we hope that this review could also be of interest to theoretical ecologists because many potentially interesting results are buried in the vast NT literature. We propose to make these more accessible by extracting them and presenting them in a logical fashion. We conclude the review by discussing how one might introduce realistic non-neutral elements into the current models., Comment: 55 pages, 8 figures

Neutral dynamics, where taxa are assumed to be demographically equivalent and their abundance is governed solely by the stochasticity of the underlying birth-death process, has proved itself as an important minimal model that accounts for many empirical datasets in genetics and ecology. However, the restriction of the model to demographic [${\cal{O}} ({\sqrt N})$] noise yields relatively slow dynamics that appears to be in conflict with both short-term and long-term characteristics of the observed systems. Here we analyze two of these problems - age size relationships and species extinction time - in the framework of a neutral theory with both demographic and environmental stochasticity. It turns out that environmentally induced variations of the demographic rates control the long-term dynamics and modify dramatically the predictions of the neutral theory with demographic noise only, yielding much better agreement with empirical data. We consider two prototypes of "zero mean" environmental noise, one which is balanced with regard to the arithmetic abundance, another balanced in the logarithmic (fitness) space, study their species lifetime statistics and discuss their relevance to realistic models of community dynamics.

Understanding the assembly of ecosystems to estimate the number of species at different spatial scales is a challenging problem. Until now, maximum entropy approaches have lacked the important feature of considering space in an explicit manner. We propose a spatially explicit maximum entropy model suitable to describe spatial patterns such as the species area relationship and the endemic area relationship. Starting from the minimal information extracted from presence/absence data, we compare the behavior of two models considering the occurrence or lack thereof of each species and information on spatial correlations. Our approach uses the information at shorter spatial scales to infer the spatial organization at larger ones. We also hypothesize a possible ecological interpretation of the effective interaction we use to characterize spatial clustering.

Social ecological systems are often difficult to investigate and manage because of their inherent complexity1. Small variations in external drivers can lead to abrupt changes associated with instabilities and bifurcations in the underlying dynamics2-4. Anticipating critical transitions and divergence from the present state of the system is particularly crucial to the prevention or mitigation of the effects of unwanted and irreversible changes5-10. Recent research in ecology has focused on leading indicators of regime shift in ecosystems characterized by one state variable5,7,11,12. The case of systems with several mutually interacting components, however, has remained poorly investigated13, while the connection between network stability and research on indicators for loss of resilience has been elusive14. Here we develop a theoretical framework to analyze early warning signs of instability and regime shift in social ecological networks. We provide analytical expressions for a set of precursors of instability in social ecological systems with additive noise for a variety of network structures. In particular, we show that the covariance matrix of the dynamics can effectively anticipate the emergence of instability. We also compare signals of early warning based on the dynamics of suitably selected nodes, to indicators based on the integrated behavior of the whole network. We find that the performances of these indicators are affected by the network structure and the type of interaction among nodes. These results provide new advances in multidimensional early warning analysis and offer a framework to evaluate the resilience of social ecological networks., Comment: 14 pages, 4 figures. Supplementary Information available upon request

Mutualistic networks are formed when the interactions between two classes of species are mutually beneficial. They are important examples of cooperation shaped by evolution. Mutualism between animals and plants plays a key role in the organization of ecological communities. Such networks in ecology have generically evolved a nested architecture independent of species composition and latitude - specialists interact with proper subsets of the nodes with whom generalists interact. Despite sustained efforts to explain observed network structure on the basis of community-level stability or persistence, such correlative studies have reached minimal consensus. Here we demonstrate that nested interaction networks could emerge as a consequence of an optimization principle aimed at maximizing the species abundance in mutualistic communities. Using analytical and numerical approaches, we show that because of the mutualistic interactions, an increase in abundance of a given species results in a corresponding increase in the total number of individuals in the community, as also the nestedness of the interaction matrix. Indeed, the species abundances and the nestedness of the interaction matrix are correlated by an amount that depends on the strength of the mutualistic interactions. Nestedness and the observed spontaneous emergence of generalist and specialist species occur for several dynamical implementations of the variational principle under stationary conditions. Optimized networks, while remaining stable, tend to be less resilient than their counterparts with randomly assigned interactions. In particular, we analytically show that the abundance of the rarest species is directly linked to the resilience of the community. Our work provides a unifying framework for studying the emergent structural and dynamical properties of ecological mutualistic networks., Comment: 10 pages, 4 figures

Empirical evidence suggesting that living systems might operate in the vicinity of critical points, at the borderline between order and disorder, has proliferated in recent years, with examples ranging from spontaneous brain activity to flock dynamics. However, a well-founded theory for understanding how and why interacting living systems could dynamically tune themselves to be poised in the vicinity of a critical point is lacking. Here we employ tools from statistical mechanics and information theory to show that complex adaptive or evolutionary systems can be much more efficient in coping with diverse heterogeneous environmental conditions when operating at criticality. Analytical as well as computational evolutionary and adaptive models vividly illustrate that a community of such systems dynamically self-tunes close to a critical state as the complexity of the environment increases while they remain non-critical for simple and predictable environments. A more robust convergence to criticality emerges in co-evolutionary and co-adaptive set-ups in which individuals aim to represent other agents in the community with fidelity, thereby creating a collective critical ensemble and providing the best possible trade-off between accuracy and flexibility. Our approach provides a parsimonious and general mechanism for the emergence of critical-like behavior in living systems needing to cope with complex environments or trying to efficiently coordinate themselves as an ensemble., Comment: Main text: 9 pages, 3 figures. Supplementary Information: 22 pages, 14 figures, 3 videos

Modern ecology has re-emphasized the need for a quantitative understanding of the original 'survival of the fittest theme' based on analyzis of the intricate trade-offs between competing evolutionary strategies that characterize the evolution of life. This is key to the understanding of species coexistence and ecosystem diversity under the omnipresent constraint of limited resources. In this work we propose an agent based model replicating a community of interacting individuals, e.g. plants in a forest, where all are competing for the same finite amount of resources and each competitor is characterized by a specific growth-reproduction strategy. We show that such an evolution dynamics drives the system towards a stationary state characterized by an emergent optimal strategy, which in turn depends on the amount of available resources the ecosystem can rely on. We find that the share of resources used by individuals is power-law distributed with an exponent directly related to the optimal strategy. The model can be further generalized to devise optimal strategies in social and economical interacting systems dynamics., Comment: 10 pages, 5 figures

Population growth is in general constrained by food production, which in turn depends on the access to water resources. At a country level, some populations use more water than they control because of their ability to import food and the virtual water required for its production. Here, we investigate the dependence of demographic growth on available water resources for exporting and importing nations. By quantifying the carrying capacity of nations based on calculations of the virtual water available through the food trade network, we point to the existence of a global water unbalance. We suggest that current export rates will not be maintained and consequently we question the long-run sustainability of the food trade system as a whole. Water rich regions are likely to soon reduce the amount of virtual water they export, thus leaving import-dependent regions without enough water to sustain their populations. We also investigate the potential impact of possible scenarios that might mitigate these effects through (1) cooperative interactions among nations whereby water rich countries maintain a tiny fraction of their food production available for export; (2) changes in consumption patterns; and (3) a positive feedback between demographic growth and technological innovations. We find that these strategies may indeed reduce the vulnerability of water-controlled societies., Comment: 11 pages, 3 figures

In the last years, a remarkable theoretical effort has been made in order to understand stability and complexity in ecological communities. The non-random structures of real ecological interaction networks has been recognized as one key ingredient contributing to the coexistence between high complexity and stability in real ecosystems. However most of the theoretical studies have considered communities with only one interaction type (either antagonistic, competitive, or mutualistic). Recently it has been proposed a theoretical analysis on multiple interaction types in ecological systems, concluding that: a) Mixture of antagonistic and mutualistic interactions stabilize the system with respect to the less realistic case of only one interaction type; b) Complexity, measured in terms of the number of interactions and the number of species, increases stability in systems with different types of interactions. By introducing new theoretical investigations and analyzing 21 empirical data sets representing mutualistic plant-pollinator networks, we show that that conclusions are incorrect. We will prove that the positive complexity-stability effect observed in systems with different kind of interactions is a mere consequence of a rescaling of the interaction strengths, and thus unrelated to the mixing of interaction types., Comment: 12 pages, 5 figures

We present novel analytical results about ecosystem species diversity that stem from a proposed coarse grained neutral model based on birth-death processes. The relevance of the problem lies in the urgency for understanding and synthesizing both theoretical results of ecological neutral theory and empirical evidence on species diversity preservation. Neutral model of biodiversity deals with ecosystems in the same trophic level where per-capita vital rates are assumed to be species-independent. Close-form analytical solutions for neutral theory are obtained within a coarse-grained model, where the only input is the species persistence time distribution. Our results pertain: the probability distribution function of the number of species in the ecosystem both in transient and stationary states; the n-points connected time correlation function; and the survival probability, definned as the distribution of time-spans to local extinction for a species randomly sampled from the community. Analytical predictions are also tested on empirical data from a estuarine fish ecosystem. We find that emerging properties of the ecosystem are very robust and do not depend on specific details of the model, with implications on biodiversity and conservation biology., Comment: 20 pages, 4 figures. To appear in Journal of Statistichal Mechanics

Natural ecosystems are characterized by striking diversity of form and functions and yet exhibit deep symmetries emerging across scales of space, time and organizational complexity. Species-area relationships and species-abundance distributions are examples of emerging patterns irrespective of the details of the underlying ecosystem functions. Here we present empirical and theoretical evidence for a new macroecological pattern related to the distributions of local species persistence times, defined as the timespans between local colonizations and extinctions in a given geographic region. Empirical distributions pertaining to two different taxa, breeding birds and herbaceous plants, analyzed in a new framework that accounts for the finiteness of the observational period, exhibit power-law scaling limited by a cut-off determined by the rate of emergence of new species. In spite of the differences between taxa and spatial scales of analysis, the scaling exponents are statistically indistinguishable from each other and significantly different from those predicted by existing models. We theoretically investigate how the scaling features depend on the structure of the spatial interaction network and show that the empirical scaling exponents are reproduced once a two-dimensional isotropic texture is used, regardless of the details of the ecological interactions. The framework developed here also allows to link the cut-off timescale with the spatial scale of analysis, and the persistence-time distribution to the species-area relationship. We conclude that the inherent coherence obtained between spatial and temporal macroecological patterns points at a seemingly general feature of the dynamical evolution of ecosystems., Comment: 5 pages, 5 figures. Supplementary materials avaliable on http://www.pnas.org/content/108/11/4346

Generalized Langevin equations (GLE) with multiplicative white Poisson noise pose the usual prescription dilemma leading to different evolution equations (master equations) for the probability distribution. Contrary to the case of multiplicative gaussian white noise, the Stratonovich prescription does not correspond to the well known mid-point (or any other intermediate) prescription. By introducing an inertial term in the GLE we show that the Ito and Stratonovich prescriptions naturally arise depending on two time scales, the one induced by the inertial term and the other determined by the jump event. We also show that when the multiplicative noise is linear in the random variable one prescription can be made equivalent to the other by a suitable transformation in the jump probability distribution. We apply these results to a recently proposed stochastic model describing the dynamics of primary soil salinization, in which the salt mass balance within the soil root zone requires the analysis of different prescriptions arising from the resulting stochastic differential equation forced by multiplicative white Poisson noise whose features are tailored to the characters of the daily precipitation. A method is finally suggested to infer the most appropriate prescription from the data.

Higher temperatures expected by midcentury increase the risk of shocks to crop production, while the interconnected nature of the current global food system functions to spread the impact of localized production shocks throughout the world. In this study, we analyze the global potential impact of a present-day event of equivalent magnitude to the US Dust Bowl, modeling the ways in which a sudden decline in US wheat production could cascade through the global network of agricultural trade. We use observations of country-level production, reserves, and trade data in a Food Shock Cascade model to explore trade adjustments and country-level inventory changes in response to a major, multiyear production decline. We find that a 4-year decline in wheat production of the same proportional magnitude as occurred during the Dust Bowl greatly reduces both wheat supply and reserves in the United States and propagates through the global trade network. By year 4 of the event, US wheat exports fall from 90.5 trillion kcal before the drought to 48 trillion to 52 trillion kcal, and the United States exhausts 94% of its reserves. As a result of reduced US exports, other countries meet their needs by leveraging their own reserves, leading to a 31% decline in wheat reserves globally. These findings demonstrate that an extreme production decline would lead to substantial supply shortfalls in both the United States and in other countries, where impacts outside the United States strongly depend on a country's reserves and on its relative position in the global trade network.