Generalized Lotka-Volterra Equations Research Articles

Heteroclinic networks for brain dynamics.

Heteroclinic networks are a mathematical concept in dynamic systems theory that is suited to describe metastable states and switching events in brain dynamics. The framework is sensitive to external input and, at the same time, reproducible and robust against perturbations. Solutions of the corresponding differential equations are spatiotemporal patterns that are supposed to encode information both in space and time coordinates. We focus on the concept of winnerless competition as realized in generalized Lotka-Volterra equations and report on results for binding and chunking dynamics, synchronization on spatial grids, and entrainment to heteroclinic motion. We summarize proposals of how to design heteroclinic networks as desired in view of reproducing experimental observations from neuronal networks and discuss the subtle role of noise. The review is on a phenomenological level with possible applications to brain dynamics, while we refer to the literature for a rigorous mathematical treatment. We conclude with promising perspectives for future research.

Quenched complexity of equilibria for asymmetric generalized Lotka–Volterra equations

We consider the generalized Lotka–Volterra system of equations with all-to-all, random asymmetric interactions describing high-dimensional, very diverse and well-mixed ecosystems. We analyse the multiple equilibria phase of the model and compute its quenched complexity, i.e. the expected value of the logarithm of the number of equilibria of the dynamical equations. We discuss the resulting distribution of equilibria as a function of their diversity, stability and average abundance. We obtain the quenched complexity by means of the replicated Kac–Rice formalism, and compare the results with the same quantity obtained within the annealed approximation, as well as with the results of the cavity calculation and, in the limit of symmetric interactions, of standard methods to compute the complexity developed in the context of glasses.

Generalized Lotka-Volterra Equations with Random, Nonreciprocal Interactions: The Typical Number of Equilibria.

We compute the typical number of equilibria of the generalized Lotka-Volterra equations describing species-rich ecosystems with random, nonreciprocal interactions using the replicated Kac-Rice method. We characterize the multiple-equilibria phase by determining the average abundance and similarity between equilibria as a function of their diversity (i.e., of the number of coexisting species) and of the variability of the interactions. We show that linearly unstable equilibria are dominant, and that the typical number of equilibria differs with respect to the average number.

Breakdown of Random-Matrix Universality in Persistent Lotka-Volterra Communities.

The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this Letter, we offer a pertinent counterexample in the context of the generalized Lotka-Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. We thus show that the eigenvalue spectra of random matrices can be used to deduce the stability of "feasible" ecological communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account.

Complex ecological communities and the emergence of island species-area relationships

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. The clustering of same-species individuals is central to our results, which we incorporate by modifying the self-interaction term in the generalized Lotka-Volterra equations. Our analysis demonstrates that the two most widely reported relationship forms can emerge due to differences in immigration rates and skewness towards weak interactions. We particularly highlight the incidence of the semi-log SAR for low immigration rates from a source pool, which is consistent with several previous empirical studies. The two SAR forms might show good fits to data over a large span of areas but a power-law overestimates species richness on smaller islands in remote archipelagoes.

Modeling the bacterial dynamics in the gut microbiota following an antibiotic-induced perturbation.

Recent studies have highlighted the importance of ecological interactions in dysbiosis of gut microbiota, but few focused on their role in antibiotic‐induced perturbations. We used the data from the CEREMI trial in which 22 healthy volunteers received a 3‐day course of ceftriaxone or cefotaxime antibiotics. Fecal samples were analyzed by 16S rRNA gene profiling, and the total bacterial counts were determined in each sample by flux cytometry. As the gut exposure to antibiotics could not be experimentally measured despite a marked impact on the gut microbiota, it was reconstructed using the counts of susceptible Escherichia coli. The dynamics of absolute counts of bacterial families were analyzed using a generalized Lotka–Volterra equations and nonlinear mixed effect modeling. Bacterial interactions were studied using a stepwise approach. Two negative and three positive interactions were identified. Introducing bacterial interactions in the modeling approach better fitted the data, and provided different estimates of antibiotic effects on each bacterial family than a simple model without interaction. The time to return to 95% of the baseline counts was significantly longer in ceftriaxone‐treated individuals than in cefotaxime‐treated subjects for two bacterial families: Akkermansiaceae (median [range]: 11.3 days [0; 180.0] vs. 4.2 days [0; 25.6], p = 0.027) and Tannerellaceae (13.7 days [6.1; 180.0] vs. 6.2 days [5.4; 17.3], p = 0.003). Taking bacterial interaction as well as individual antibiotic exposure profile into account improves the analysis of antibiotic‐induced dysbiosis.

Multi-stable bacterial communities exhibit extreme sensitivity to initial conditions.

Microbial communities can have dramatically different compositions even among similar environments. This might be due to the existence of multiple alternative stable states, yet there exists little experimental evidence supporting this possibility. Here, we gathered a large collection of absolute population abundances capturing population dynamics in one- to four-strain communities of soil bacteria with a complex life cycle in a feast-or-famine environment. This dataset led to several observations: (i) some pairwise competitions resulted in bistability with a separatrix near a 1:1 initial ratio across a range of population densities; (ii) bistability propagated to multi-stability in multispecies communities; and (iii) replicate microbial communities reached different stable states when starting close to initial conditions separating basins of attraction, indicating finite-sized regions where the dynamics are unpredictable. The generalized Lotka-Volterra equations qualitatively captured most competition outcomes but were unable to quantitatively recapitulate the observed dynamics. This was partly due to complex and diverse growth dynamics in monocultures that ranged from Allee effects to nonmonotonic behaviors. Overall, our results highlight that multi-stability might be generic in multispecies communities and, combined with ecological noise, can lead to unpredictable community assembly, even in simple environments.

Exact Reduction of the Generalized Lotka–Volterra Equations via Integral and Algebraic Substitutions

Systems of interacting species, such as biological environments or chemical reactions, are often described mathematically by sets of coupled ordinary differential equations. While a large number β of species may be involved in the coupled dynamics, often only α<β species are of interest or of consequence. In this paper, we explored how to construct models that include only those given α species, but still recreate the dynamics of the original β-species model. Under some conditions detailed here, this reduction can be completed exactly, such that the information in the reduced model is exactly the same as the original one, but over fewer equations. Moreover, this reduction process suggests a promising type of approximate model—no longer exact, but computationally quite simple.

Making quantitative predictions on the yield of a species immersed in a multispecies community: The focal species method

• Predicting the species yields of a community is central for its management. • Estimating all the required parameters to attain such goal may be unfeasible. • The focal species approximation requires only a fraction of these parameters. • Predictions are compared against experiments covering a wide variety of taxa. • Predictions using only a fraction of the set of model parameters are quite accurate. The abundance or yield of a species is an ecological quantity of paramount importance when making management and conservation decisions. The linear generalized Lotka–Volterra equations (LGLVE) constitute the simplest theoretical framework for predicting such yields. However, in many ecological communities ( e.g. tropical forests), the number of coexisting species S can be very large and estimating the entire interaction matrix A from experiments is practically unfeasible. Here, I introduce a method for predicting the yield of a particular species of interest, or focal species. The method proposed here only requires data on the interaction coefficients of the focal species k , referring to the elements of row and column k of the approximate focal interaction matrix A ( k ) . These interaction coefficients can be obtained from measuring the yields of species in monoculture ( S experimental treatments) and in biculture experiments involving the focal species ( S − 1 treatments). The remaining elements of A ( k ) are considered equal to − 1. I compare the species yields predicted by the focal species method against a data set of experimental studies I compiled from the literature across several taxa −plants, algae, protozoa and crustacean. The accuracy of this method is slightly lower than that attained when using the entire Lotka-Volterra interaction matrix. The advantage of this method is that it requires a much smaller number of experimental treatments to estimate parameters (of the order of S rather than of S 2 ). Therefore, it can be a useful quantitative tool for applied sciences like agriculture or fisheries.

Ecological communities from random generalized Lotka-Volterra dynamics with nonlinear feedback.

We investigate the outcome of generalized Lotka-Volterra dynamics of ecological communities with random interaction coefficients and nonlinear feedback. We show in simulations that the saturation of nonlinear feedback stabilizes the dynamics. This is confirmed in an analytical generating-functional approach to generalized 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 behavior. We investigate in detail the combined effects of the mean, variance, and covariance of the random interaction coefficients, and the saturation value of the nonlinear response. We find that stability and diversity increases with the introduction of nonlinear feedback, where decreasing the saturation value has a similar effect to decreasing the covariance. We also find cooperation to no longer have a detrimental effect on stability with nonlinear feedback, and the order parameters mean abundance and diversity to be less dependent on the symmetry of interactions with stronger saturation.

When one stock share is a biological individual: a stylized simulation of the population dynamics in an order-driven market

The demand–supply relationship plays an important role in an order-driven stock market. In this paper, we propose a stylized model by defining demand (supply) over a stock at a certain time as how many shares are on the bid (ask) side, which includes all buy (sell) limit orders and buy (sell) market orders. Also, we will treat the two types of shares as two different species with interaction (a single share corresponds to an individual of one species) and will construct and apply generalized Lotka–Volterra equations (Hofbauer and Sigmund in Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998) to simulate how their population evolve based on some properties or assumptions of an order-driven market, and also on the heterogenous beliefs among traders. The model suggests that the population of bid and ask shares moves either to a fixed point or periodically without the impact of external information. Also, our model gives a reason, though it is not perfect, explaining why stock prices can behave chaotically.

Bifurcation analysis of the dynamics of interacting subnetworks of a spiking network

We analyze the collective dynamics of hierarchically structured networks of densely connected spiking neurons. These networks of sub-networks may represent interactions between cell assemblies or different nuclei in the brain. The dynamical activity pattern that results from these interactions depends on the strength of synaptic coupling between them. Importantly, the overall dynamics of a brain region in the absence of external input, so called ongoing brain activity, has been attributed to the dynamics of such interactions. In our study, two different network scenarios are considered: a system with one inhibitory and two excitatory subnetworks, and a network representation with three inhibitory subnetworks. To study the effect of synaptic strength on the global dynamics of the network, two parameters for relative couplings between these subnetworks are considered. For each case, a bifurcation analysis is performed and the results have been compared to large-scale network simulations. Our analysis shows that Generalized Lotka-Volterra (GLV) equations, well-known in predator-prey studies, yield a meaningful population-level description for the collective behavior of spiking neuronal interaction, which have a hierarchical structure. In particular, we observed a striking equivalence between the bifurcation diagrams of spiking neuronal networks and their corresponding GLV equations. This study gives new insight on the behavior of neuronal assemblies, and can potentially suggest new mechanisms for altering the dynamical patterns of spiking networks based on changing the synaptic strength between some groups of neurons.

Predicting the separation of time scales in a heteroclinic network

Abstract We consider a heteroclinic network in the framework of winnerless competition, realized by generalized Lotka-Volterra equations. By an appropriate choice of predation rates we impose a structural hierarchy so that the network consists of a heteroclinic cycle of three heteroclinic cycles which connect saddles on the basic level. As we have demonstrated in previous work, the structural hierarchy can induce a hierarchy in time scales such that slow oscillations modulate fast oscillations of species concentrations. Here we derive a Poincaré map to determine analytically the number of revolutions of the trajectory within one heteroclinic cycle on the basic level, before it switches to the heteroclinic connection on the second level. This provides an understanding of which parameters control the separation of time scales and determine the decisions of the trajectory at branching points of this network.

On predicting species yields in multispecies communities: Quantifying the accuracy of the linear Lotka-Volterra generalized model

The linear generalized Lotka–Volterra equations (LGLVE) constitute the simplest theoretical framework for ecological communities involving different kinds of interspecific interactions ―e.g. competition, facilitation. These equations have been often criticized as being too simple to model real systems.This study has two main goals: First, to test the LGLVE as a quantitative tool for describing/explaining/predicting the equilibrium species abundances. That is, how accurately the LGLVE predict the yields of S interacting species? Second, to show analyze which quantitative predictions are possible with an incomplete knowledge of the LGLVE parameters.With this aim I compiled from the literature 33 experiments, most of them for plants, which measured species yields in monoculture, biculture and in mixtures of S >2 species.I found that, by obtaining the LGLVE parameters from the yields in monoculture and biculture experiments, the LGLVE can accurately predict the majority of the equilibrium species yields in the mixtures of S >2 species for most of the experiments.However, in many natural communities, e.g. tropical forests or plankton, S can be of the order of hundreds and estimating all the model parameters from empirical data is unfeasible. But, by estimating the mean interaction coefficient from the yields of an incomplete set of monoculture and biculture experiments, the LGLVE still can make accurate predictions. Firstly, it is possible to derive simple formulas which are able to predict the relative yield total and mean relative yield. Secondly, when we are interested in the fate or performance of a particular species, we can use a more refined approximation to predict the yield of this focal species with accuracy comparable to the one obtained when using the full set of LGLVE.

Theory of invasion extinction dynamics in minimal food webs.

When food webs are exposed to species invasion, secondary extinction cascades may be set off. Although much work has gone into characterizing the structure of food webs, systematic predictions on their evolutionary dynamics are still scarce. Here we present a theoretical framework that predicts extinctions in terms of an alternating sequence of two basic processes: resource depletion by or competitive exclusion between consumers. We first propose a conceptual invasion extinction model (IEM) involving random fitness coefficients. We bolster this IEM by an analytical, recursive procedure for calculating idealized extinction cascades after any species addition and simulate the long-time evolution. Our procedure describes minimal food webs where each species interacts with only a single resource through the generalized Lotka-Volterra equations. For such food webs ex- tinction cascades are determined uniquely and the system always relaxes to a stable steady state. The dynamics and scale invariant species life time resemble the behavior of the IEM, and correctly predict an upper limit for trophic levels as observed in the field.

Ecological communities with Lotka-Volterra dynamics.

Ecological communities in heterogeneous environments assemble through the combined effect of species interaction and migration. Understanding the effect of these processes on the community properties is central to ecology. Here we study these processes for a single community subject to migration from a pool of species, with population dynamics described by the generalized Lotka-Volterra equations. We derive exact results for the phase diagram describing the dynamical behaviors, and for the diversity and species abundance distributions. A phase transition is found from a phase where a unique globally attractive fixed point exists to a phase where multiple dynamical attractors exist, leading to history-dependent community properties. The model is shown to possess a symmetry that also establishes a connection with other well-known models.

Food Web Assembly Rules for Generalized Lotka-Volterra Equations.

In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. For the generalized Lotka-Volterra equations, sustainable coexistence necessitates nonzero determinant of the interaction matrix. Here we show that this requirement is equivalent to demanding that each species be part of a non-overlapping pairing, which substantially constrains the food web structure. We demonstrate that a stable food web can always be obtained if a non-overlapping pairing exists. If it does not, the matrix rank can be used to quantify the lack of niches, corresponding to unpaired species. For the species richness at each trophic level, we derive the food web assembly rules, which specify sustainable combinations. In neighboring levels, these rules allow the higher level to avert competitive exclusion at the lower, thereby incorporating apparent competition. In agreement with data, the assembly rules predict high species numbers at intermediate levels and thinning at the top and bottom. Using comprehensive food web data, we demonstrate how omnivores or parasites with hosts at multiple trophic levels can loosen the constraints and help obtain coexistence in food webs. Hence, omnivory may be the glue that keeps communities intact even under extinction or ecological release of species.

Noise Induced Phenomena in the Dynamics of Two Competing Species

Noise through its interaction with the nonlinearity of the living systems can give rise to counter-intuitive phenomena. In this paper we shortly review noise induced effects in different ecosystems, in which two populations compete for the same resources. We also present new results on spatial patterns of two populations, while modeling real distributions of anchovies and sardines. The transient dynamics of these ecosystems are analyzed through generalized Lotka-Volterra equations in the presence of multiplicative noise, which models the interaction between the species and the environment. We find noise induced phenomena such as quasi-deterministic oscillations, stochastic resonance, noise delayed extinction, and noise induced pattern formation. In addition, our theoretical results are validated with experimental findings. Specifically the results, obtained by a coupled map lattice model, well reproduce the spatial distributions of anchovies and sardines, observed in a marine ecosystem. Moreover, the experimental dynamical behavior of two competing bacterial populations in a meat product and the probability distribution at long times of one of them are well reproduced by a stochastic microbial predictive model.

Sequential memory: Binding dynamics.

Temporal order memories are critical for everyday animal and human functioning. Experiments and our own experience show that the binding or association of various features of an event together and the maintaining of multimodality events in sequential order are the key components of any sequential memories-episodic, semantic, working, etc. We study a robustness of binding sequential dynamics based on our previously introduced model in the form of generalized Lotka-Volterra equations. In the phase space of the model, there exists a multi-dimensional binding heteroclinic network consisting of saddle equilibrium points and heteroclinic trajectories joining them. We prove here the robustness of the binding sequential dynamics, i.e., the feasibility phenomenon for coupled heteroclinic networks: for each collection of successive heteroclinic trajectories inside the unified networks, there is an open set of initial points such that the trajectory going through each of them follows the prescribed collection staying in a small neighborhood of it. We show also that the symbolic complexity function of the system restricted to this neighborhood is a polynomial of degree L - 1, where L is the number of modalities.

Two competing species in super-diffusive dynamical regimes

The dynamics of two competing species within the framework of the generalized Lotka-Volterra equations, in the presence of multiplicative α-stable Levy noise sources and a random time dependent interaction parameter, is studied. The species dynamics is characterized by two different dynamical regimes, exclusion of one species and coexistence of both, depending on the values of the interaction parameter, which obeys a Langevin equation with a periodically fluctuating bistable potential and an additive α-stable Levy noise. The stochastic resonance phenomenon is analyzed for noise sources asymmetrically distributed. Finally, the effects of statistical dependence between multiplicative noise and additive noise on the dynamics of the two species are studied.