Generalized Lotka-Volterra Model Research Articles

Heterogeneous mean-field analysis of the generalized Lotka–Volterra model on a network

Abstract We study the dynamics of the generalized Lotka–Volterra model with a network structure. Performing a high connectivity expansion for graphs, we write down a mean-field dynamical theory that incorporates degree heterogeneity. This allows us to describe the fixed points of the model in terms of a few simple order parameters. We extend the analysis even for diverging abundances, using a mapping to the replicator model. With this we present a unified approach for both cooperative and competitive systems that display complementary behaviors. In particular we show the central role of an order parameter called the critical degree, g c . In the competitive regime g c serves to distinguish high degree nodes that are more likely to go extinct, while in the cooperative regime it has the reverse role, it will determine the low degree nodes that tend to go relatively extinct.

Alternative cliques of coexisting species in complex ecosystems

The possibility that some ecosystems can exist in alternative stable states has profound implications for ecosystem conservation and restoration. Current ecological theory on multistability mostly relies on few-species dynamical models, in which alternative states are intrinsically related to specific non-linear dynamics. Recent theoretical advances, however, have shown that multiple stable ‘cliques’—small subsets of coexisting species—can be present in species-rich models even under linear interactions. Yet, the mechanisms governing the appearence and characteristics of these cliques remain largely unexplored. In the present work, we investigate cliques in the generalized Lotka–Volterra model with mathematical and computational techniques. Our findings reveal that simple probabilistic and dynamical constraints can explain the appearence, properties and stability of cliques. Our work contributes to the understanding of alternative stable states in complex ecological communities.

Sediment DNA Records the Critical Transition of Bacterial Communities in the Arid Lake

It is necessary to predict the critical transition of lake ecosystems due to their abrupt, non-linear effects on social-economic systems. Given the promising application of paleolimnological archives to tracking the historical changes of lake ecosystems, it is speculated that they can also record the lake’s critical transition. We studied Lake Dali-Nor in the arid region of Inner Mongolia because of the profound shrinking the lake experienced between the 1300 s and the 1600 s. We reconstructed the succession of bacterial communities from a 140-cm-long sediment core at 4-cm intervals and detected the critical transition. Our results showed that the historical trajectory of bacterial communities from the 1200 s to the 2010s was divided into two alternative states: state1 from 1200 to 1300 s and state2 from 1400 to 2010s. Furthermore, in the late 1300 s, the appearance of a tipping point and critical slowing down implied the existence of a critical transition. By using a multi-decadal time series from the sedimentary core, with general Lotka-Volterra model simulations, local stability analysis found that bacterial communities were the most unstable as they approached the critical transition, suggesting that the collapse of stability triggers the community shift from an equilibrium state to another state. Furthermore, the most unstable community harbored the strongest antagonistic and mutualistic interactions, which may imply the detrimental role of interaction strength on community stability. Collectively, our study showed that sediment DNA can be used to detect the critical transition of lake ecosystems.

Kinetics-based inference of environment-dependent microbial interactions and their dynamic variation.

Microbial communities in nature are dynamically evolving as member species change their interactions subject to environmental variations. Accounting for such context-dependent dynamic variations in interspecies interactions is critical for predictive ecological modeling. In the absence of generalizable theoretical foundations, we lack a fundamental understanding of how microbial interactions are driven by environmental factors, significantly limiting our capability to predict and engineer community dynamics and function. To address this issue, we propose a novel theoretical framework that allows us to represent interspecies interactions as an explicit function of environmental variables (such as substrate concentrations) by combining growth kinetics and a generalized Lotka-Volterra model. A synergistic integration of these two complementary models leads to the prediction of alterations in interspecies interactions as the outcome of dynamic balances between positive and negative influences of microbial species in mixed relationships. The effectiveness of our method was experimentally demonstrated using a synthetic consortium of two Escherichia coli mutants that are metabolically dependent (due to an inability to synthesize essential amino acids) but competitively grow on a shared substrate. The analysis of the E. coli binary consortium using our model not only showed how interactions between the two amino acid auxotrophic mutants are controlled by the dynamic shifts in limiting substrates but also enabled quantifying previously uncharacterizable complex aspects of microbial interactions, such as asymmetry in interactions. Our approach can be extended to other ecological systems to model their environment-dependent interspecies interactions from growth kinetics.IMPORTANCEModeling environment-controlled interspecies interactions through separate identification of positive and negative influences of microbes in mixed relationships is a new capability that can significantly improve our ability to understand, predict, and engineer the complex dynamics of microbial communities. Moreover, the prediction of microbial interactions as a function of environmental variables can serve as valuable benchmark data to validate modeling and network inference tools in microbial ecology, the development of which has often been impeded due to the lack of ground truth information on interactions. While demonstrated against microbial data, the theory developed in this work is readily applicable to general community ecology to predict interactions among macroorganisms, such as plants and animals, as well as microorganisms.

High immigration rates critical for establishing emigration-driven diversity in microbial communities

Unraveling the mechanisms governing the diversity of ecological communities is a central goal in ecology. Although microbial dispersal constitutes an important ecological process, the effect of dispersal on microbial diversity is poorly understood. Here, we sought to fill this gap by combining a generalized Lotka-Volterra model with experimental investigations. Our model showed that emigration increases the diversity of the community when the immigration rate crosses a defined threshold, which we identified as Ineutral. We also found that at high immigration rates, emigration weakens the relative abundance of fast-growing species and thus enhances the mass effect and increases the diversity. We experimentally confirmed this finding using co-cultures of 20 bacterial strains isolated from the soil. Our model further showed that Ineutral decreases with the increase of species pool size, growth rate, and interspecies interaction. Our work deepens the understanding of the effects of dispersal on the diversity of natural communities.

Metabolic model-based ecological modeling for probiotic design.

The microbial community composition in the human gut has a profound effect on human health. This observation has lead to extensive use of microbiome therapies, including over-the-counter 'probiotic' treatments intended to alter the composition of the microbiome. Despite so much promise and commercial interest, the factors that contribute to the success or failure of microbiome-targeted treatments remain unclear. We investigate the biotic interactions that lead to successful engraftment of a novel bacterial strain introduced to the microbiome as in probiotic treatments. We use pairwise genome-scale metabolic modeling with a generalized resource allocation constraint to build a network of interactions between taxa that appear in an experimental engraftment study. We create induced sub-graphs using the taxa present in individual samples and assess the likelihood of invader engraftment based on network structure. To do so, we use a generalized Lotka-Volterra model, which we show has strong ability to predict if a particular invader or probiotic will successfully engraft into an individual's microbiome. Furthermore, we show that the mechanistic nature of the model is useful for revealing which microbe-microbe interactions potentially drive engraftment.

Geometric optimal control of the generalized Lotka–Volterra model of the intestinal microbiome

AbstractWe introduce the theoretical framework from geometric optimal control for a control system modeled by the generalized Lotka–Volterra equation, motivated by restoring the gut microbiota infected by Clostridium difficile combining antibiotic treatment and fecal injection. We consider both permanent control and sampled‐data control related to the medical protocols.

When cheating turns into a stabilizing mechanism of plant-pollinator communities.

Mutualistic interactions, such as plant-mycorrhizal or plant-pollinator interactions, are widespread in ecological communities and frequently exploited by cheaters, species that profit from interactions without providing benefits in return. Cheating usually negatively affects the fitness of the individuals that are cheated on, but the effects of cheating at the community level remains poorly understood. Here, we describe 2 different kinds of cheating in mutualistic networks and use a generalized Lotka-Volterra model to show that they have very different consequences for the persistence of the community. Conservative cheating, where a species cheats on its mutualistic partners to escape the cost of mutualistic interactions, negatively affects community persistence. In contrast, innovative cheating occurs with species with whom legitimate interactions are not possible, because of a physiological or morphological barrier. Innovative cheating can enhance community persistence under some conditions: when cheaters have few mutualistic partners, cheat at low or intermediate frequency and the cost associated with mutualism is not too high. Under these conditions, the negative effects of cheating on partner persistence are overcompensated at the community level by the positive feedback loops that arise in diverse mutualistic communities. Using an empirical dataset of plant-bird interactions (hummingbirds and flowerpiercers), we found that observed cheating patterns are highly consistent with theoretical cheating patterns found to increase community persistence. This result suggests that the cheating patterns observed in nature could contribute to promote species coexistence in mutualistic communities, instead of necessarily destabilizing them.

Cyclic pairwise interaction representing a rock-paper-scissors game maintains the population of the vulnerable yeast Saccharomyces cerevisiae within a multispecies sourdough microbiome.

Sourdough starters are spontaneously generated multispecies communities consisting of lactic acid bacteria (LAB) and yeasts. Traditionally used to make baked goods, these communities of microorganisms can be propagated through successive transfers to other containers. Successive transfers result in microbial community dynamics; however, the mechanisms guiding the dynamics are not fully elucidated. This study tracked five species of sourdough LAB-yeast communities (three LAB species and two yeast species) during successive transfers in vitro to explore the interspecies interactions driving the multispecies community dynamics. The microbial dynamics were modeled with a generalized Lotka-Volterra (gLV) equation, which can mathematically identify ecological types of interspecies interactions. The gLV model simulated the population dynamics of each species with Pearson's correlation coefficient values of ≥0.817. Pairwise experiments revealed that competition (negative/negative) predominated among all pairwise interactions (70%), followed by amensalism (negative/neutral) (30%). In the pairwise community, the LAB species Limosilactobacillus fermentum suppressed Saccharomyces cerevisiae growth through amensalism; however, S. cerevisiae population was maintained when the community comprised 3-5 species, including the LAB species L. fermentum and Pediococcus pentosaceus. The key interspecies interactions for maintaining the S. cerevisiae population were theoretically determined using the gLV model; the three identified species interacted non-transitively with S. cerevisiae in a cyclic pairwise interaction, metaphorically representing a rock-paper-scissors relationship, which systemically supported the S. cerevisiae population in the multispecies community. These theoretical insights indicate that cyclic pairwise interaction is the main driver of LAB-yeast population dynamics, which helps sustain a vulnerable yeast species in a sourdough multispecies community.IMPORTANCETraditionally, multispecies consisting of lactic acid bacteria and yeasts collaboratively engage sourdough fermentation, which determines the quality of the resulting baked goods. Nonetheless, the successive transfer of these microbial communities can result in undesirable community dynamics that prevent the formation of high-quality sourdough bread. Thus, a mechanistic understanding of the community dynamics is fundamental to engineer sourdough complex fermentation. This study describes the population dynamics of five species of lactic acid bacteria-yeast communities in vitro using a generalized Lotka-Volterra model that examines interspecies interactions. A vulnerable yeast species was maintained within up to five species community dynamics by obtaining support with a cyclic interspecies interaction. Metaphorically, it involves a rock-paper-scissors game between two lactic acid bacteria species. Application of the generalized Lotka-Volterra model to real food microbiomes including sourdoughs will increase the reliability of the model prediction and help identify key microbial interactions that drive microbiome dynamics.

Plant Community Prediction Based on Lotka-Volterra Model and Statistical Analysis

The change process of plant communities is an important issue involving botany, environmental science and system science. This paper mainly studies the state changes of plant communities under irregular weather cycles, and establishes a dynamic model of species interaction based on Lotka-Volterra equations. The Euler method, statistical analysis are used to solve the problem. we extend the basic Lotka-Volterra model to the generalized Lotka-Volterra model, which can describe the change of a community with multiple species. Based on this, by introducing parameters such as drought intensity, species sensitivity to drought, pollution level and habitat reduction into the equations, a model for predicting the change of plant community status under the influence of irregular weather cycles is obtained. The statistical method is creatively used to do Sensitivity analysis for the model. The results show that: (1) pollution and habitat reduction do not affect the advantages of highly drought-resistant plants, but extend the gain range of species number on plant community stability. (2) the influence of some negative factors on the gain interval is the comprehensive result of the influence of two opposite factors, competition intensity and species diversity.

Emergent non-Hermitian physics in a generalized Lotka-Volterra model

Emergent non-Hermitian physics in a generalized Lotka-Volterra model

Therapeutic interventions alter ecological interactions among cystic fibrosis airway microbiota.

The airways of people with cystic fibrosis (CF) often harbor a diverse microbiota and in recent years, much effort has been invested in cataloguing these. In spite of providing a wealth of insight, this cataloguing tells us little about how the organisms interact with one another in the CF airways. However, such relationships can be inferred using the theoretical framework of the Lotka-Volterra (LV) model. In the current work, we use a generalized Lotka-Volterra model to interrogate the nationwide data collected and curated by the UK CF Registry. This longitudinal dataset (covering the period 2008-2020) contains annual depositions that record the presence/absence of microbial taxa in each patient, their medication, and their CF genotype. Specifically, we wanted to identify trends in ecological relationships between the CF microbiota at a nationwide level, and whether these are potentially affected by medication. Our results show that some medications have a distinct influence on the microbial interactome, especially those that potentially influence the "gut-lung axis" or mucus viscosity. In particular, we found that patients treated with a combination of antimicrobial agents (targeting the airway microbiota), digestive enzymes (assisting in the assimilation of dietary fats and carbohydrates), and DNase (to reduce mucus viscosity) displayed a distinctly different airway interactome compared with patients treated separately with these medications.

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.

Bioeconomic analysis of harvesting within a predator–prey system: A case study in the Chesapeake Bay fisheries

Sustainable use of biological resources is very important as over exploitation on the long run may lead to stock depletion, which in turn may threaten biodiversity. The Chesapeake Bay is an extremely complex ecosystem, and sustainable harvesting of its fisheries is essential both for the ecosystem’s biodiversity and economic prosperity of the area. Here, we use ecosystem based mathematical modeling to study the population dynamics with harvesting of two key fishes in the Chesapeake Bay, the Atlantic Menhaden (Brevoortia tyrannus) as a prey and the Striped Bass (Morone saxatilis) as a predator. We start by fitting the generalized Lotka–Volterra model to actual time series abundance data of the two species obtained from fisheries in the Bay. We derive conditions for the existence of the bio-economic equilibrium and investigate the stability and the resilience of the biological system. We study the maximum sustainable yield, maximum economic yield, and resilience maximizing yield policies and their effects on the fisheries long term sustainability, particularly with respect to the menhaden-bass population dynamics. This study may be used by policy-makers to balance the economic and ecological harvesting goals while managing the populations of Atlantic menhaden and striped bass in the Chesapeake Bay fisheries.

Proportional stochastic generalized Lotka–Volterra model with an application to learning microbial community structures

Inferring microbial community structure based on temporal metagenomics data is an important goal in microbiome studies. The deterministic generalized Lotka–Volterra (GLV) differential equations have been commonly used to model the dynamics of microbial taxa. However, these approaches fail to take random environmental fluctuations into account and usually ignore the compositional nature of relative abundance data, which may deteriorate the estimates. In this article, we consider the microbial dynamics in terms of relative abundances by introducing a reference taxon, and propose a new proportional stochastic GLV (pSGLV) differential equation model, where the random perturbations of Brownian motion in this model can naturally account for the external environmental effects on the microbial community. We establish conditions and show some mathematical properties of the solutions including general existence and uniqueness, stochastic ultimate boundedness, stochastic permanence, the existence of stationary distribution, and ergodicity property. We further develop approximate maximum likelihood estimators (AMLEs) based on discrete observations and systematically investigate the consistency and asymptotic normality of the proposed estimators. At last, numerical simulations support our theoretical findings and our method is demonstrated through an application to the well-known “moving picture” temporal microbial dataset.

Metabolic models of human gut microbiota: Advances and challenges.

The human gut is a complex ecosystem consisting of hundreds of microbial species interacting with each other and with the human host. Mathematical models of the gut microbiome integrate our knowledge of this system and help to formulate hypotheses to explain observations. The generalized Lotka-Volterra model has been widely used for this purpose, but it does not describe interaction mechanisms and thus does not account for metabolic flexibility. Recently, models that explicitly describe gut microbial metabolite production and consumption have become popular. These models have been used to investigate the factors that shape gut microbial composition and to link specific gut microorganisms to changes in metabolite concentrations found in diseases. Here, we review how such models are built and what we have learned so far from their application to human gut microbiome data. In addition, we discuss current challenges of these models and how these can be addressed in the future.

Analysis of the dynamical behavior of solutions for a class of hybrid generalized Lotka–Volterra models

In the paper, the generalized Lotka–Volterra systems with switching are investigated. It is assumed that functions included in the right parts of the considered systems satisfy nonlinear constraints of the power type. Conditions are found for the switching law that provide the desired features of the dynamical behavior of solutions of such systems. In particular, the problems to guarantee the asymptotic stability of a given equilibrium position with the required region of attraction, the ultimate boundedness of solutions, the global asymptotic stability are studied. To solve these problems, the method of splitting the phase space of the studied systems into several parts is used. In each of these parts, some partial estimates for the chosen Lyapunov function are constructed and conditions are established to ensure the transition of solutions from one part to another. Moreover, different Lyapunov functions can be used in different parts of the phase space.

Species abundance correlations carry limited information about microbial network interactions.

Unraveling the network of interactions in ecological communities is a daunting task. Common methods to infer interspecific interactions from cross-sectional data are based on co-occurrence measures. For instance, interactions in the human microbiome are often inferred from correlations between the abundances of bacterial phylogenetic groups across subjects. We tested whether such correlation-based methods are indeed reliable for inferring interaction networks. For this purpose, we simulated bacterial communities by means of the generalized Lotka-Volterra model, with variation in model parameters representing variability among hosts. Our results show that correlations can be indicative for presence of bacterial interactions, but only when measurement noise is low relative to the variation in interaction strengths between hosts. Indication of interaction was affected by type of interaction network, process noise and sampling under non-equilibrium conditions. The sign of a correlation mostly coincided with the nature of the strongest pairwise interaction, but this is not necessarily the case. For instance, under rare conditions of identical interaction strength, we found that competitive and exploitative interactions can result in positive as well as negative correlations. Thus, cross-sectional abundance data carry limited information on specific interaction types. Correlations in abundance may hint at interactions but require independent validation.

Chemical-Mediated Microbial Interactions Can Reduce the Effectiveness of Time-Series-Based Inference of Ecological Interaction Networks.

Network-based assessments are important for disentangling complex microbial and microbial–host interactions and can provide the basis for microbial engineering. There is a growing recognition that chemical-mediated interactions are important for the coexistence of microbial species. However, so far, the methods used to infer microbial interactions have been validated with models assuming direct species-species interactions, such as generalized Lotka–Volterra models. Therefore, it is unclear how effective existing approaches are in detecting chemical-mediated interactions. In this paper, we used time series of simulated microbial dynamics to benchmark five major/state-of-the-art methods. We found that only two methods (CCM and LIMITS) were capable of detecting interactions. While LIMITS performed better than CCM, it was less robust to the presence of chemical-mediated interactions, and the presence of trophic competition was essential for the interactions to be detectable. We show that the existence of chemical-mediated interactions among microbial species poses a new challenge to overcome for the development of a network-based understanding of microbiomes and their interactions with hosts and the environment.

Effects of intraspecific cooperative interactions in large ecosystems

We analyze the role of the Allee effect - a positive correlation between population density and mean individual fitness - for ecological communities formed by a large number of species. Our study is performed using the generalized Lotka-Volterra model with random interactions between species. We obtain the phase diagram and analyze the nature of the multiple equilibria phase. Remarkable differences emerge with respect to the logistic growth case, thus revealing the major role played by the functional response in determining aggregate behaviors of large ecosystems.