We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors.
We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (Journal of Mathematical Biology '19) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Annals of Applied Probability '18) are strengthened to include degenerate noise and auxiliary variables.
Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.
Ecologists are interested in understanding and predicting how ecological communities change through time. While it might seem natural to measure this through changes in species' abundances, computational limitations mean transitions between community types are often modelled instead. We present an approach inspired by DNA substitution models that attempts to estimate historic interactions between species, and thus estimate turnover rates in ecological communities. Although our simulations show that the method has some limitations, our application to butterfly community data shows the method ca…
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 stoch…
Having control over species abundances and community resilience is of great interest for experimental, agricultural, industrial and conservation purposes. Here, we theoretically explore the possibility of manipulating ecological communities by modifying pairwise interactions. Specifically, we establish which interaction values should be modified, and by how much, in order to alter the composition or resilience of a community towards a favorable direction. While doing so, we also take into account the experimental difficulties in making such modifications by including in our optimization proces…
Emerging research suggests that the extent to which activity spaces -- the collection of an individual's routine activity locations -- overlap provides important information about the functioning of a city and its neighborhoods. To study patterns of overlapping activity spaces, we draw on the notion of an ecological network, a type of two-mode network with the two modes being individuals and the geographic locations where individuals perform routine activities. We describe a method for detecting "ecological communities" within these networks based on shared activity locations among individuals…
The far-reaching consequences of ecological interactions in the dynamics of biological communities remain an intriguing subject. For decades, competition has been a cornerstone in ecological processes, but mounting evidence shows that cooperation does also contribute to the structure of biological communities. Here, we propose a simple deterministic model for the study of the effects of facilitation and competition in the dynamics of such systems. The simultaneous inclusion of both effects produces rich dynamics and captures the context-dependence observed in the formation of ecological commun…
Emergent frequency-dependent selection predicts mutation outcomes in complex ecological communities
MIT Center for Theoretical Physics - a Leinweber Institute（麻省理工学院理论物理中心（莱因韦伯研究所））；Department of Physics, Boston University（波士顿大学物理系）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究院国际理论科学中心）；Faculty of Computing and Data Sciences, Boston University（波士顿大学计算与数据科学学院）
In 1972, Robert May triggered a worldwide research program studying ecological communities using random matrix theory. Yet, it remains unclear if and when we can treat real communities as random ecosystems. Here, we draw on recent progress in random matrix theory and statistical physics to extend May's approach to generalized consumer-resource models. We show that in diverse ecosystems adding even modest amounts of noise to consumer preferences results in a transition to "typicality" where macroscopic ecological properties of communities are indistinguishable from those of random ecosystems, e…
The assembly of ecological communities from a pool of species is central to ecology, but the effect of this process on properties of community interaction networks is still largely unknown. Here, we use a systematic analytical framework to describe how assembly from a species pool gives rise to community network properties that differ from those of the pool: Compared to the pool, the community shows a bias towards higher carrying capacities, weaker competitive interactions and stronger beneficial interactions. Moreover, even if interactions between all pool species are completely random, commu…
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 stoch…
Assembling optimal microbial communities is key for various applications in biofuel production, agriculture, and human health. Finding the optimal community is challenging because the number of possible communities grows exponentially with the number of species, and so an exhaustive search cannot be performed even for a dozen species. A heuristic search that improves community function by adding or removing one species at a time is more practical, but it is unknown whether this strategy can discover an optimal or nearly optimal community. Using consumer-resource models with and without cross-f…
Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among $m$ ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for $m=2$ no coexistence of the two species is usually possible, if $m \ge 3$ we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coex…
It has been a century since the species-area relationship (SAR) was first proposed as a power law to explain how species richness scales with area. There have been many attempts to explain the origin of this predominant form. Apart from the power law, numerous empirical studies also report a semi-log form of the SAR, but very few have addressed its incidence. In this work, we test whether these relationships could emerge from the assembly of large random communities on island-like systems. We reformulate the generalized Lotka-Volterra model by introducing an area parameter that determines the …
Minimal model of self-organized clusters with phase transitions in ecological communities
MIT Center for Theoretical Physics - a Leinweber Institute（麻省理工学院理论物理中心-莱因韦伯研究所）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）；Department of Physics, Boston University（波士顿大学物理系）
The origin of diversification and coexistence of genes and species have been traditionally studied in isolated biological levels. Ecological and evolutionary views have focused on the mechanisms that enable or constrain species coexistence, genetic variation and the genetics of speciation, but a unified theory linking those approaches is still missing. Here we introduce evolutionary graphs in the context of neutral theories of molecular evolution and biodiversity to provide a framework that simultaneously addresses speciation rate and joint genetic and species diversities. To illuminate this q…
We investigate the outcome of generalised Lotka-Volterra dynamics of ecological communities with random interaction coefficients and non-linear feedback. We show in simulations that the saturation of non-linear feedback stabilises the dynamics. This is confirmed in an analytical generating-functional approach to generalised Lotka-Volterra equations with piecewise linear saturating response. For such systems we are able to derive self-consistent relations governing the stable fixed-point phase, and to carry out a linear stability analysis to predict the onset of unstable behaviour. We investiga…
Ecosystems often undergo abrupt regime shifts in response to gradual external changes. These shifts are theoretically understood as a regime switch between alternative stable states of the ecosystem dynamical response to smooth changes in external conditions. Usual models introduce nonlinearities in the macroscopic dynamics of the ecosystem that lead to different stable attractors among which the shift takes place. Here we propose an alternative explanation of catastrophic regime shifts based on a recent model that pictures ecological communities as systems in continuous fluctuation, according…
Ecologists have put forward many explanations for coexistence, but these are only partial explanations; nature is complex, so it is reasonable to assume that in any given ecological community, multiple mechanisms of coexistence are operating at the same time. Here, we present a methodology for quantifying the relative importance of different explanations for coexistence, based on an extension of Modern Coexistence Theory. Current versions of Modern Coexistence Theory only allow for the analysis of communities that are affected by spatial or temporal environmental variation, but not both. We sh…
We present properties of Lotka-Volterra equations describing ecological competition among a large number of competing species. First we extend to the case of a non-homogeneous niche space stability conditions for solutions representing species coexistence. Second, we discuss mechanisms leading to species clustering and obtain an analytical solution for a lumped state in a specific instance of the system. We also discuss how realistic ecological interactions may result in different types of competition coefficients.
The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. We prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapuno…
Statistical Physics has proved essential to analyze multi-agent environments. Motivated by the empirical observation of various non-equilibrium features in Barro Colorado and other ecological systems, we analyze a plant-species abundance model, presenting analytical evidence of scale-invariant plant clusters and non-trivial emergent modular correlations. Such first theoretical confirmation of a scale-invariant region, based on percolation processes, reproduces the key features in actual ecological ecosystems and can confer the most stable equilibrium for ecosystems with vast biodiversity.
To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When…
Species experience both internal feedbacks with endogenous factors such as trait evolution and external feedbacks with exogenous factors such as weather. These feedbacks can play an important role in determining whether populations persist or communities of species coexist. To provide a general mathematical framework for studying these effects, we develop a theorem for coexistence for ecological models accounting for internal and external feedbacks. Specifically, we use average Lyapunov functions and Morse decompositions to develop sufficient and necessary conditions for robust permanence, a f…
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 mor…
In order to understand the role of space in ecological communities where each species produces a certain type of resource and has varying abilities to exploit the resources produced by its own species and by the other species, we carry out a comparative study of an interacting particle system and its mean-field approximation. For a wide range of parameter values, we show both analytically and numerically that the spatial model results in predictions that significantly differ from its nonspatial counterpart, indicating that the use of the mean-field approach to describe the evolution of communi…
Ecological networks are theoretical abstractions that represent ecological communities. These networks are usually defined as static entities, in which the occurrence of a particular interaction between species is considered fixed despite the intrinsic dynamics of ecological systems. However, empirical analysis of the temporal variation of trophic interactions is constrained by the lack of data with high spatial, temporal, and taxonomic resolution. Here, we evaluate the spatiotemporal variability of multiple consumer-resource interactions of large marine networks. The tropic interactions of al…
In plant-pollinator communities many pollinators are potential generalists and their preferences for certain plants can change quickly in response to changes in plant and pollinator densities. These changes in preferences affect coexistence within pollinator guilds as well as within plant guilds. Using a mathematical model, we study how adaptations of pollinator preferences influence population dynamics of a two-plant-two-pollinator community interaction module. Adaptation leads to coexistence between generalist and specialist pollinators, and produces complex plant population dynamics, involv…
The dynamics of many microbial ecosystems are driven by cross-feeding interactions, in which metabolites excreted by some species are metabolised further by others. The population dynamics of such ecosystems are governed by frequency-dependent selection, which allows for stable coexistence of two or more species. We have analysed a model of cross-feeding based on the replicator equation, with the aim of establishing criteria for coexistence in ecosystems containing three species, given the information of the three species' ability to coexist in their three separate pairs, i.e. the long term dy…
The dynamics of ecosystem collapse are fundamental to determining how and why biological communities change through time, as well as the potential effects of extinctions on ecosystems. Here we integrate depictions of mammals from Egyptian antiquity with direct lines of paleontological and archeological evidence to infer local extinctions and community dynamics over a 6000-year span. The unprecedented temporal resolution of this data set enables examination of how the tandem effects of human population growth and climate change can disrupt mammalian communities. We show that the extinctions of …
An ongoing debate in ecology concerns the impacts of ecological drift and selection on community assembly. Here, we show that there is a sharp phase transition in diverse ecological communities between a selection dominated regime (the niche phase) and a drift dominated regime (the neutral phase). Simulations and analytic arguments show that the niche phase is favored in communities with large population sizes and relatively constant environments, whereas the neutral phase is favored in communities with small population sizes and fluctuating environments. Our results demonstrate how apparently…
Coexistence coalitions in propagule disperser quasi-communities
Centre d’Estudis Avançats de Blanes, Consejo Superior de Investigaciones Científicas (CEAB-CSIC)（布兰斯高级研究中心，西班牙国家研究委员会）；Universidad Politécnica de Madrid（马德里理工大学）
Diverse communities promote the coexistence of closely-related strains through emergent equalization and stabilization
National Centre for Biological Sciences, Tata Institute of Fundamental Research（塔塔基础科学研究所国家生物科学中心）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础科学研究所国际理论科学中心）
Evolutionary and ecosystem dynamics are often treated as different processes --operating at separate timescales-- even if evidence reveals that rapid evolutionary changes can feed back into ecological interactions. A recent long-term field experiment has explicitly shown that communities of competing plant species can experience very fast phenotypic diversification, and that this gives rise to enhanced complementarity in resource exploitation and to enlarged ecosystem-level productivity. Here, we build on progress made in recent years in the integration of eco-evolutionary dynamics, and presen…
Quantitative predictions about the processes that promote species coexistence are a subject of active research in ecology. In particular, competitive interactions are known to shape and maintain ecological communities, and situations where some species out-compete or dominate over some others are key to describe natural ecosystems. Here we develop ecological theory using a stochastic, synthetic framework for plant community assembly that leads to predictions amenable to empirical testing. We propose two stochastic continuous-time Markov models that incorporate competitive dominance through a h…
Current theories about mechanisms promoting species co-existence in diverse communities assume that species only interact ecologically. Species are treated as discrete evolutionary entities, even though abundant empirical evidence indicates that patterns of gene flow such as selfing and hybridization frequently occur in plant and animal groups. Here, we allow mating behavior to respond to local species composition and abundance in a data-driven meta-community model of species co-existence. While individuals primarily out-cross, they also maintain some small capacity for selfing and hybridizati…
Dispersal of species is a fundamental ecological process in the evolution and maintenance of biodiversity. Limited control over ecological parameters has hindered progress in understanding of what enables species to colonise new area, as well as the importance of inter-species interactions. Such control is necessary to construct reliable mathematical models of ecosystems. In our work, we studied dispersal in the context of bacterial range expansions and identified the major determinants of species coexistence for a bacterial model system of three Escherichia coli strains (toxin producing, sens…
Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires th…
