Extinction in neutrally stable stochastic Lotka-Volterra models

Populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions. The processes leading to extinction of species, whose understanding is a key component in the study of evolution and biodiversity, are influenced by both of these factors. In this paper, we investigate a class of stochastic population dynamics models based on generalized Lotka-Volterra systems. In the case of neutral stability of the underlying deterministic model, the impact of intrinsic noise on the survival of species is dramatic: it destroys coexistence of interacting species on a time scale proportional to the population size. We introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower-dimensional effective dynamics. This is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations. The extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations.Comment: 14 pages, 7 figure

Avalanche shape and exponents beyond mean-field theory

Elastic systems, such as magnetic domain walls, density waves, contact lines, and cracks, are all pinned by substrate disorder. When driven, they move via successive jumps called avalanches, with power law distributions of size, duration and velocity. Their exponents, and the shape of an avalanche, defined as its mean velocity as function of time, have recently been studied. They are known approximatively from experiments and simulations, and were predicted from mean-field models, such as the Brownian force model (BFM), where each point of the elastic interface sees a force field which itself is a random walk. As we showed in EPL 97 (2012) 46004, the BFM is the starting point for an $\epsilon = d_{\rm c}-d$ expansion around the upper critical dimension, with $d_{\rm c}=4$ for short-ranged elasticity, and $d_{\rm c}=2$ for long-ranged elasticity. Here we calculate analytically the ${\cal O}(\epsilon)$, i.e. 1-loop, correction to the avalanche shape at fixed duration $T$, for both types of elasticity. The exact expression is well approximated by \left_T\simeq [ Tx(1-x)]^{\gamma-1} \exp\left( {\cal A}\left[\frac12-x\right]\right), $0<x<1$. The asymmetry ${\cal A}\approx - 0.336 (1-d/d_{\rm c})$ is negative for $d$ close to $d_{\rm c}$, skewing the avalanche towards its end, as observed in numerical simulations in $d=2$ and $3$. The exponent $\gamma=(d+\zeta)/z$ is given by the two independent exponents at depinning, the roughness $\zeta$ and the dynamical exponent $z$. We propose a general procedure to predict other avalanche exponents in terms of $\zeta$ and $z$. We finally introduce and calculate the shape at fixed avalanche size, not yet measured in experiments or simulations.Comment: 6 pages, 2 figure

Statistics of Avalanches with Relaxation, and Barkhausen Noise: A Solvable Model

We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model of a particle in a Brownian force landscape, including retardation effects. We show that under monotonous driving the particle moves forward at all times, as it does in absence of retardation (Middleton's theorem). This remarkable property allows us to develop an analytical treatment. The model with an exponentially decaying memory kernel is realized in Barkhausen experiments with eddy-current relaxation, and has previously been shown numerically to account for the experimentally observed asymmetry of Barkhausen-pulse shapes. We elucidate another qualitatively new feature: the breakup of each avalanche of the standard ABBM model into a cluster of sub-avalanches, sharply delimited for slow relaxation under quasi-static driving. These conditions are typical for earthquake dynamics. With relaxation and aftershock clustering, the present model includes important ingredients for an effective description of earthquakes. We analyze quantitatively the limits of slow and fast relaxation for stationary driving with velocity v>0. The v-dependent power-law exponent for small velocities, and the critical driving velocity at which the particle velocity never vanishes, are modified. We also analyze non-stationary avalanches following a step in the driving magnetic field. Analytically, we obtain the mean avalanche shape at fixed size, the duration distribution of the first sub-avalanche, and the time dependence of the mean velocity. We propose to study these observables in experiments, allowing to directly measure the shape of the memory kernel, and to trace eddy current relaxation in Barkhausen noise.Comment: 39 pages, 26 figure

Mobility-dependent selection of competing strategy associations

Standard models of population dynamics focus on the interaction, survival, and extinction of the competing species individually. Real ecological systems, however, are characterized by an abundance of species (or strategies, in the terminology of evolutionary-game theory) that form intricate, complex interaction networks. The description of the ensuing dynamics may be aided by studying associations of certain strategies rather than individual ones. Here we show how such a higher-level description can bear fruitful insight. Motivated from different strains of colicinogenic Escherichia coli bacteria, we investigate a four-strategy system which contains a three-strategy cycle and a neutral alliance of two strategies. We find that the stochastic, spatial model exhibits a mobility-dependent selection of either the three-strategy cycle or of the neutral pair. We analyze this intriguing phenomenon numerically and analytically.Peer reviewe

Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model

We obtain an exact solution for the motion of a particle driven by a spring in a Brownian random-force landscape, the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model. Many experiments on quasi-static driving of elastic interfaces (Barkhausen noise in magnets, earthquake statistics, shear dynamics of granular matter) exhibit the same universal behavior as this model. It also appears as a limit in the field theory of elastic manifolds. Here we discuss predictions of the ABBM model for monotonous, but otherwise arbitrary, time-dependent driving. Our main result is an explicit formula for the generating functional of particle velocities and positions. We apply this to derive the particle-velocity distribution following a quench in the driving velocity. We also obtain the joint avalanche size and duration distribution and the mean avalanche shape following a jump in the position of the confining spring. Such non-stationary driving is easy to realize in experiments, and provides a way to test the ABBM model beyond the stationary, quasi-static regime. We study extensions to two elastically coupled layers, and to an elastic interface of internal dimension d, in the Brownian force landscape. The effective action of the field theory is equal to the action, up to 1-loop corrections obtained exactly from a functional determinant. This provides a connection to renormalization-group methods.Comment: 18 pages, 3 figure

Distribution of velocities in an avalanche

For a driven elastic object near depinning, we derive from first principles the distribution of instantaneous velocities in an avalanche. We prove that above the upper critical dimension, d >= d_uc, the n-times distribution of the center-of-mass velocity is equivalent to the prediction from the ABBM stochastic equation. Our method allows to compute space and time dependence from an instanton equation. We extend the calculation beyond mean field, to lowest order in epsilon=d_uc-d.Comment: 4 pages, 2 figure

Transient fluctuation of the prosperity of firms in a network economy

The transient fluctuation of the prosperity of firms in a network economy is investigated with an abstract stochastic model. The model describes the profit which firms make when they sell materials to a firm which produces a product and the fixed cost expense to the firms to produce those materials and product. The formulae for this model are parallel to those for population dynamics. The swinging changes in the fluctuation in the transient state from the initial growth to the final steady state are the consequence of a topology-dependent time trial competition between the profitable interactions and expense. The firm in a sparse random network economy is more likely to go bankrupt than expected from the value of the limit of the fluctuation in the steady state, and there is a risk of failing to reach by far the less fluctuating steady state

On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols

In this work we focus on a natural class of population protocols whose dynamics are modelled by the discrete version of Lotka-Volterra equations. In such protocols, when an agent $a$ of type (species) $i$ interacts with an agent $b$ of type (species) $j$ with $a$ as the initiator, then $b$'s type becomes $i$ with probability $P\_{ij}$. In such an interaction, we think of $a$ as the predator, $b$ as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an $n$-agent population converges to some absorbing state in time polynomial in $n$, w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We start by presenting a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but "control" convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a distributed system. Some of our techniques may be of independent value

Interference in disordered systems: A particle in a complex random landscape

We consider a particle in one dimension submitted to amplitude and phase disorder. It can be mapped onto the complex Burgers equation, and provides a toy model for problems with interplay of interferences and disorder, such as the NSS model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. The model has three distinct phases: (I) a {\em high-temperature} or weak disorder phase, (II) a {\em pinned} phase for strong amplitude disorder, and (III) a {\em diffusive} phase for strong phase disorder, but weak amplitude disorder. We compute analytically the renormalized disorder correlator, equivalent to the Burgers velocity-velocity correlator at long times. In phase III, it assumes a universal form. For strong phase disorder, interference leads to a logarithmic singularity, related to zeroes of the partition sum, or poles of the complex Burgers velocity field. These results are valuable in the search for the adequate field theory for higher-dimensional systems.Comment: 16 pages, 7 figure