Phase transition and 1/f noise in a game dynamical model

The persistence of bipartite ecological communities with Lotka-Volterra dynamics

The assembly and persistence of ecological communities can be understood as the result of the interaction and migration of species. Here we study a single community subject to migration from a species pool in which inter-specific interactions are organised according to a bipartite network. Considering the dynamics of species abundances to be governed by generalised Lotka-Volterra equations, we extend work on unipartite networks to we derive exact results for the phase diagram of this model. Focusing on antagonistic interactions, we describe factors that influence the persistence of the two guilds, locate transitions to multiple-attractor and unbounded phases, as well as identifying a region of parameter space in which consumers are essentially absent in the local community.

Local and extensive fluctuations in sparsely interacting ecological communities

Ecological communities with many species can be classified into dynamical phases. In systems with all-to-all interactions, a phase where species abundances always reach a fixed point and a phase where they continuously fluctuate have been found. The dynamics when interactions are sparse, with each species interacting with only a few others, has remained largely unexplored. Here we study a system of sparse interactions, first when interactions are of constant strength and completely unidirectional, and then when adding variability and bidirectionality. We show that in this case a phase unique to the sparse setting appears in the phase diagram, where for the same control parameters different communities may reach either a fixed point or a state where the abundances of only a finite subset of species fluctuate, and we calculate the probability for each outcome. These fluctuating species are organized around short cycles in the interaction graph, and their abundances undergo large nonlinear fluctuations. We characterize the approach from this phase to a phase with extensively many fluctuating species, and show that the probability of fluctuations grows continuously to one as the transition is approached, and that the number of fluctuating species diverges. This is qualitatively distinct from the transition to extensive fluctuations coming from a fixed point phase, which is marked by a loss of linear stability. The differences are traced back to the emergent binary character of the dynamics when far from short cycles.

Niche overlap and Hopfield-like interactions in generalized random Lotka-Volterra systems

We study communities emerging from generalized random Lotka-Volterra dynamics with a large number of species with interactions determined by the degree of niche overlap. Each species is endowed with a number of traits, and competition between pairs of species increases with their similarity in trait space. This leads to a model with random Hopfield-like interactions. We use tools from the theory of disordered systems, notably dynamic mean-field theory, to characterize the statistics of the resulting communities at stable fixed points and determine analytically when stability breaks down. Two distinct types of transition are identified in this way, both marked by diverging abundances but differing in the behavior of the integrated response function. At fixed points only a fraction of the initial pool of species survives. We numerically study the eigenvalue spectra of the interaction matrix between extant species. We find evidence that the two types of dynamical transition are, respectively, associated with the bulk spectrum or an outlier eigenvalue crossing into the right half of the complex plane.

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.

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

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

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.

Generalized Lotka-Volterra model with hierarchical interactions

In the analysis of complex ecosystems it is common to use random interaction coefficients, which are often assumed to be such that all species are statistically equivalent. In this work we relax this assumption by imposing hierarchical interspecies interactions. These are incorporated into a generalized Lotka-Volterra dynamical system. In a hierarchical community species benefit more, on average, from interactions with species further below them in the hierarchy than from interactions with those above. Using dynamic mean-field theory, we demonstrate that a strong hierarchical structure is stabilizing, but that it reduces the number of species in the surviving community, as well as their abundances. Additionally, we show that increased heterogeneity in the variances of the interaction coefficients across positions in the hierarchy is destabilizing. We also comment on the structure of the surviving community and demonstrate that the abundance and probability of survival of a species are dependent on its position in the hierarchy.

Parliamentary roll-call voting as a complex dynamical system: The case of Chile

A method is proposed to study the temporal variability of legislative roll-call votes in a parliament from the perspective of complex dynamical systems. We studied the Chilean Chamber of Deputies' by analyzing the agreement ratio and the voting outcome of each vote over the last 19 years with a Recurrence Quantification Analysis and an entropy analysis (Sample Entropy). Two significant changes in the temporal variability were found: one in 2014, where the voting outcome became more recurrent and with less entropy, and another in 2018, where the agreement ratio became less recurrent and with higher entropy. These changes may be directly related to major changes in the Chilean electoral system and the composition of the Chamber of Deputies, given that these changes occurred just after the first parliamentary elections with non-compulsory voting (2013 elections) and the first elections with a proportional system in conjunction with an increase in the number of deputies (2017 elections) were held.

Emergent phases of ecological diversity and dynamics mapped in microcosms

From tropical forests to gut microbiomes, ecological communities host notably high numbers of coexisting species. Beyond high biodiversity, communities exhibit a range of complex dynamics that are difficult to explain under a unified framework. Using bacterial microcosms, we performed a direct test of theory predicting that simple community-level features dictate emergent behaviors of communities. As either the number of species or the strength of interactions increases, we show that microbial ecosystems transition between three distinct dynamical phases, from a stable equilibrium in which all species coexist to partial coexistence to emergence of persistent fluctuations in species abundances, in the order predicted by theory. Under fixed conditions, high biodiversity and fluctuations reinforce each other. Our results demonstrate predictable emergent patterns of diversity and dynamics in ecological communities.

Gradient dynamics in reinforcement learning

Despite the success achieved by the analysis of supervised learning algorithms in the framework of statistical mechanics, reinforcement learning has remained largely untouched by physicists. Here we move towards closing the gap by analyzing the dynamics of the policy gradient algorithm. For a convex problem, namely the k-armed bandit, we show that the learning dynamics obeys a drift-diffusion motion described by a Langevin equation, the coefficients of which can be tuned by the learning rate. We explore the striking similarity between our Langevin equation and the Kimura equation, describing genotypes evolution. Furthermore, we propose a mapping between a nonconvex reinforcement learning setting describing multiple joints of a robotic arm and a disordered system, namely a p-spin glass. This mapping enables us to show how the learning rate acts as an effective temperature and thus is capable of smoothing rough landscapes, corroborating what is displayed by the drift-diffusive description and paving the way for physics-inspired algorithmic optimization based on annealing procedures in disordered systems.

Local and collective transitions in sparsely-interacting ecological communities

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

Eigenvalues of Random Matrices with Generalized Correlations: A Path Integral Approach

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this Letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalized correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.

Consensus, polarization, and coexistence in a continuous opinion dynamics model with quenched disorder

A model of opinion dynamics is introduced in which each individual's opinion is measured on a bounded continuous spectrum. Each opinion is influenced heterogeneously by every other opinion in the population. It is demonstrated that consensus, polarization and a spread of moderate opinions are all possible within this model. Using dynamic mean-field theory, we are able to identify the statistical features of the interactions between individuals that give rise to each of the aforementioned emergent phenomena. The nature of the transitions between each of the observed macroscopic states is also studied. It is demonstrated that heterogeneity of interactions between individuals can lead to polarization, that mostly antagonistic or contrarian interactions can promote consensus at a moderate opinion, and that mostly reinforcing interactions encourage the majority to take an extreme opinion.

Impact of colonization history on the composition of ecological systems

Observational studies of ecological systems have shown that different species compositions can arise from distinct species arrival orders during community assembly-also known as colonization history. The presence of multiple interior equilibria in the positive orthant of the state space of the population dynamics will naturally lead to history dependency of the final state. However, it is still unclear whether and under which conditions colonization history will dominate community composition in the absence of multiple interior equilibria. Here, by considering that only one species can invade at a time and there are no recurrent invasions, we show clear evidence that the colonization history can have a big impact on the composition of ecological systems even in the absence of multiple interior equilibria. In particular, we first derive two simple rules to determine whether the composition of a community will depend on its colonization history in the absence of multiple interior equilibria and recurrent invasions. Then we apply them to communities governed by generalized Lotka-Volterra (gLV) dynamics and propose a numerical scheme to measure the probability of colonization history dependence. Finally, we show, via numerical simulations, that for gLV dynamics with a single interior equilibrium, the probability that community composition is dominated by colonization history increases monotonically with community size, network connectivity, and the variation of intrinsic growth rates across species. These results reveal that in the absence of multiple interior equilibria and recurrent invasions, community composition is a probabilistic process mediated by ecological dynamics via the interspecific variation and the size of regional pools.

Dynamical persistence in high-diversity resource-consumer communities

We show how highly-diverse ecological communities may display persistent abundance fluctuations, when interacting through resource competition and subjected to migration from a species pool. These fluctuations appear, robustly and predictably, in certain regimes of parameter space. Their origin is closely tied to the ratio of realized species diversity to the number of resources. This ratio is set by competition, through the balance between species being pushed out and invading. When this ratio is smaller than one, dynamics will reach stable equilibria. When this ratio is larger than one, the competitive exclusion principle dictates that fixed-points are either unstable or marginally stable. If they are unstable, the system is repelled from fixed points, and abundances forever fluctuate. While marginally-stable fixed points are in principle allowed and predicted by some models, they become structurally unstable at high diversity. This means that even small changes to the model, such as non-linearities in how resources combine to generate species' growth, will result in persistent abundance fluctuations.

Complex interactions can create persistent fluctuations in high-diversity ecosystems

When can ecological interactions drive an entire ecosystem into a persistent non-equilibrium state, where many species populations fluctuate without going to extinction? We show that high-diversity spatially heterogeneous systems can exhibit chaotic dynamics which persist for extremely long times. We develop a theoretical framework, based on dynamical mean-field theory, to quantify the conditions under which these fluctuating states exist, and predict their properties. We uncover parallels with the persistence of externally-perturbed ecosystems, such as the role of perturbation strength, synchrony and correlation time. But uniquely to endogenous fluctuations, these properties arise from the species dynamics themselves, creating feedback loops between perturbation and response. A key result is that fluctuation amplitude and species diversity are tightly linked: in particular, fluctuations enable dramatically more species to coexist than at equilibrium in the very same system. Our findings highlight crucial differences between well-mixed and spatially-extended systems, with implications for experiments and their ability to reproduce natural dynamics. They shed light on the maintenance of biodiversity, and the strength and synchrony of fluctuations observed in natural systems.

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.

Best reply structure and equilibrium convergence in generic games

Game theory is widely used to model interacting biological and social systems. In some situations, players may converge to an equilibrium, e.g., a Nash equilibrium, but in other situations their strategic dynamics oscillate endogenously. If the system is not designed to encourage convergence, which of these two behaviors can we expect a priori? To address this question, we follow an approach that is popular in theoretical ecology to study the stability of ecosystems: We generate payoff matrices at random, subject to constraints that may represent properties of real-world games. We show that best reply cycles, basic topological structures in games, predict nonconvergence of six well-known learning algorithms that are used in biology or have support from experiments with human players. Best reply cycles are dominant in complicated and competitive games, indicating that in this case equilibrium is typically an unrealistic assumption, and one must explicitly model the dynamics of learning.

Random matrices and condensation into multiple states

In the present work, we employ methods from statistical mechanics of disordered systems to investigate static properties of condensation into multiple states in a general framework. We aim at showing how typical properties of random interaction matrices play a vital role in manifesting the statistics of condensate states. In particular, an analytical expression for the fraction of condensate states in the thermodynamic limit is provided that confirms the result of the mean number of coexisting species in a random tournament game. We also study the interplay between the condensation problem and zero-sum games with correlated random payoff matrices.

The prevalence of chaotic dynamics in games with many players

We study adaptive learning in a typical p-player game. The payoffs of the games are randomly generated and then held fixed. The strategies of the players evolve through time as the players learn. The trajectories in the strategy space display a range of qualitatively different behaviours, with attractors that include unique fixed points, multiple fixed points, limit cycles and chaos. In the limit where the game is complicated, in the sense that the players can take many possible actions, we use a generating-functional approach to establish the parameter range in which learning dynamics converge to a stable fixed point. The size of this region goes to zero as the number of players goes to infinity, suggesting that complex non-equilibrium behaviour, exemplified by chaos, is the norm for complicated games with many players.

Generic assembly patterns in complex ecological communities

The study of ecological communities often involves detailed simulations of complex networks. However, our empirical knowledge of these networks is typically incomplete and the space of simulation models and parameters is vast, leaving room for uncertainty in theoretical predictions. Here we show that a large fraction of this space of possibilities exhibits generic behaviors that are robust to modeling choices. We consider a wide array of model features, including interaction types and community structures, known to generate different dynamics for a few species. We combine these features in large simulated communities, and show that equilibrium diversity, functioning, and stability can be predicted analytically using a random model parameterized by a few statistical properties of the community. We give an ecological interpretation of this "disordered" limit where structure fails to emerge from complexity. We also demonstrate that some well-studied interaction patterns remain relevant in large ecosystems, but their impact can be encapsulated in a minimal number of additional parameters. Our approach provides a powerful framework for predicting the outcomes of ecosystem assembly and quantifying the added value of more detailed models and measurements.

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.

The Effect of Intra- and Interspecific Competition on Coexistence in Multispecies Communities

For two competing species, intraspecific competition must exceed interspecific competition for coexistence. To generalize this well-known criterion to multiple competing species, one must take into account both the distribution of interaction strengths and community structure. Here we derive a multispecies generalization of the two-species rule in the context of symmetric Lotka-Volterra competition and obtain explicit stability conditions for random competitive communities. We then explore the influence of community structure on coexistence. Results show that both the most and least stabilized cases have striking global structures, with a nested pattern emerging in both cases. The distribution of intraspecific coefficients leading to the most and least stabilized communities also follows a predictable pattern that can be justified analytically. In addition, we show that the size of the parameter space allowing for feasible communities always increases with the strength of intraspecific effects in a characteristic way that is independent of the interspecific interaction structure. We conclude by discussing possible extensions of our results to nonsymmetric competition.

Symmetry of interactions rules in incompletely connected random replicator ecosystems

The evolution of an incompletely connected system of species with speciation and extinction is investigated in terms of random replicators. It is found that evolving random replicator systems with speciation do become large and complex, depending on speciation parameters. Antisymmetric interactions result in large systems, whereas systems with symmetric interactions remain small. A co-dominating feature is within-species interaction pressure: large within-species interaction increases species diversity. Average fitness evolves in all systems, however symmetry and connectivity evolve in small systems only. Newcomers get extinct almost immediately in symmetric systems. The distribution in species lifetimes is determined for antisymmetric systems. The replicator systems investigated do not show any sign of self-organized criticality. The generalized Lotka-Volterra system is shown to be a tedious way of implementing the replicator system.

Complex dynamics in learning complicated games

Game theory is the standard tool used to model strategic interactions in evolutionary biology and social science. Traditionally, game theory studies the equilibria of simple games. However, is this useful if the game is complicated, and if not, what is? We define a complicated game as one with many possible moves, and therefore many possible payoffs conditional on those moves. We investigate two-person games in which the players learn based on a type of reinforcement learning called experience-weighted attraction (EWA). By generating games at random, we characterize the learning dynamics under EWA and show that there are three clearly separated regimes: (i) convergence to a unique fixed point, (ii) a huge multiplicity of stable fixed points, and (iii) chaotic behavior. In case (iii), the dimension of the chaotic attractors can be very high, implying that the learning dynamics are effectively random. In the chaotic regime, the total payoffs fluctuate intermittently, showing bursts of rapid change punctuated by periods of quiescence, with heavy tails similar to what is observed in fluid turbulence and financial markets. Our results suggest that, at least for some learning algorithms, there is a large parameter regime for which complicated strategic interactions generate inherently unpredictable behavior that is best described in the language of dynamical systems theory.

Phase transitions to cooperation in the prisoner's dilemma

Game theory formalizes certain interactions between physical particles or between living beings in biology, sociology, and economics and quantifies the outcomes by payoffs. The prisoner's dilemma (PD) describes situations in which it is profitable if everybody cooperates rather than defects (free rides or cheats), but as cooperation is risky and defection is tempting, the expected outcome is defection. Nevertheless, some biological and social mechanisms can support cooperation by effectively transforming the payoffs. Here, we study the related phase transitions, which can be of first order (discontinuous) or of second order (continuous), implying a variety of different routes to cooperation. After classifying the transitions into cases of equilibrium displacement, equilibrium selection, and equilibrium creation, we show that a transition to cooperation may take place even if the stationary states and the eigenvalues of the replicator equation for the PD stay unchanged. Our example is based on adaptive group pressure, which makes the payoffs dependent on the endogenous dynamics in the population. The resulting bistability can invert the expected outcome in favor of cooperation.

Rank abundance relations in evolutionary dynamics of random replicators

We present a nonequilibrium statistical mechanics description of rank abundance relations (RAR) in random community models of ecology. Specifically, we study a multispecies replicator system with quenched random interaction matrices. We here consider symmetric interactions as well as asymmetric and antisymmetric cases. RARs are obtained analytically via a generating functional analysis, describing fixed-point states of the system in terms of a small set of order parameters, and in dependence on the symmetry or otherwise of interactions and on the productivity of the community. Our work is an extension of Tokita [Phys. Rev. Lett. 93, 178102 (2004)], where the case of symmetric interactions was considered within an equilibrium setup. The species abundance distribution in our model come out as truncated normal distributions or transformations thereof and, in some case, are similar to left-skewed distributions observed in ecology. We also discuss the interaction structure of the resulting food-web of stable species at stationarity, cases of heterogeneous cooperation pressures as well as effects of finite system size and of higher-order interactions.

Controlling species richness in spin-glass model ecosystems

Within the framework of the random replicator model of ecosystems, we use equilibrium statistical mechanics tools to study the effect of manipulating the ecosystem so as to guarantee that a fixed fraction of the surviving species at equilibrium display a predefined set of characters (e.g., characters of economic value). Provided that the intraspecies competition is not too weak, we find that the consequence of such intervention on the ecosystem composition is a significant increase on the number of species that become extinct, and so the impoverishment of the ecosystem.

Mass extinction in a dynamical system of evolution with variable dimension

Introducing the effect of extinction into the so-called replicator equations in mathematical biology, we construct a general model where the diversity of species, i.e., the dimension of the equation, is a time-dependent variable. The system shows very different behavior from the original replicator equation, and leads to mass extinction when the system initially has high diversity. The present theory can serve as a mathematical foundation for the paleontologic theory for mass extinction. This extinction dynamics is a prototype of dynamical systems where the variable dimension is inevitable.