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Nicolas Lanchier

Publications and source records attributed to Nicolas Lanchier.

52 records · Page 3Linked to original sources
arXiv2013

Fixation in the one-dimensional Axelrod model

The Axelrod model is a spatial stochastic model for the dynamics of cultures which includes two important social factors: social influence, the tendency of individuals to become more similar when they interact, and homophily, the tendency of individuals to interact more frequently with individuals who are more similar. Each vertex of the interaction network is characterized by its culture, a vector of $F$ cultural features that can each assumes $q$ different states. Pairs of neighbors interact at a rate proportional to the number of cultural features they have in common, which results in the interacting pair having one more cultural feature in common. In this article, we continue the analysis of the Axelrod model initiated by the first author by proving that the one-dimensional system fixates when $F\leq cq$ where the slope satisfies the equation $e^{-c}=c$. In addition, we show that the two-feature model with at least three states fixates. This last result is sharp since it is known from previous works that the one-dimensional two-feature two-state Axelrod model clusters.

math.PR↗
arXiv2013

Fluctuation versus fixation in the one-dimensional constrained voter model

The constrained voter model describes the dynamics of opinions in a population of individuals located on a connected graph. Each agent is characterized by her opinion, where the set of opinions is represented by a finite sequence of consecutive integers, and each pair of neighbors, as defined by the edge set of the graph, interact at a constant rate. The dynamics depends on two parameters: the number of opinions denoted by $F$ and a so-called confidence threshold denoted by $θ$. If the opinion distance between two interacting agents exceeds the confidence threshold then nothing happens, otherwise one of the two agents mimics the other one just as in the classical voter model. Our main result shows that the one-dimensional system starting from any product measures with a positive density of each opinion fluctuates and clusters if and only if $F \leq 2 θ+ 1$. Sufficient conditions for fixation in one dimension when the initial distribution is uniform and lower bounds for the probability of consensus for the process on finite connected graphs are also proved.

math.PR↗
arXiv2013

Survival and extinction results for a patch model with sexual reproduction

This article is concerned with a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents' patch or are sent to a nearby patch with some fixed probabilities. Specifically, we prove lower and upper bounds for the probability of long-term survival for the process starting with a single fully occupied patch. Our main finding is that, with probability close to one and for a certain set of parameters, the metapopulation survives in the presence of nearest neighbor interactions while it dies out in the presence of long range interactions, suggesting that the best strategy for the population to expand in space is to use intermediate dispersal ranges. This result is due to the presence of a so-called strong Allee effect induced by the birth mechanism and does not hold for the basic contact process.

math.PR↗
arXiv2013

Galam's bottom-up hierarchical system and public debate model revisited

This article is concerned with the bottom-up hierarchical system and public debate model proposed by Galam, as well as a spatial version of the public debate model. In all three models, there is a population of individuals who are characterized by one of two competing opinions, say opinion -1 and opinion +1. This population is further divided into groups of common size s. In the bottom-up hierarchical system, each group elects a representative candidate, whereas in the other two models, all the members of each group discuss at random times until they reach a consensus. At each election/discussion, the winning opinion is chosen according to Galam's majority rule: the opinion with the majority of representants wins when there is a strict majority while one opinion, say opinion -1, is chosen by default in case of a tie. For the public debate models, we also consider the following natural updating rule that we shall call proportional rule: the winning opinion is chosen at random with a probability equal to the fraction of its supporters in the group. The three models differ in term of their population structure: in the bottom-up hierarchical system, individuals are located on a finite regular tree, in the non-spatial public debate model, they are located on a complete graph, and in the spatial public debate model, they are located on the $d$-dimensional regular lattice. For the bottom-up hierarchical system and non-spatial public debate model, Galam studied the probability that a given opinion wins under the majority rule and assuming that individuals' opinions are initially independent, making the initial number of supporters of a given opinion a binomial random variable. The first objective of this paper is to revisit his result assuming that the initial number of individuals in favor of a given opinion is a fixed deterministic number ...

math.PR↗
arXiv2013

Stochastic dynamics on hypergraphs and the spatial majority rule model

This article starts by introducing a new theoretical framework to model spatial systems which is obtained from the framework of interacting particle systems by replacing the traditional graphical structure that defines the network of interactions with a structure of hypergraph. This new perspective is more appropriate to define stochastic spatial processes in which large blocks of vertices may flip simultaneously, which is then applied to define a spatial version of the Galam's majority rule model. In our spatial model, each vertex of the lattice has one of two possible competing opinions, say opinion 0 and opinion 1, as in the popular voter model. Hyperedges are updated at rate one, which results in all the vertices in the hyperedge changing simultaneously their opinion to the majority opinion of the hyperedge. In the case of a tie in hyperedges with even size, a bias is introduced in favor of type 1, which is motivated by the principle of social inertia. Our analytical results along with simulations and heuristic arguments suggest that, in any spatial dimensions and when the set of hyperedges consists of the collection of all n x ... x n blocks of the lattice, opinion 1 wins when n is even while the system clusters when n is odd, which contrasts with results about the voter model in high dimensions for which opinions coexist. This is fully proved in one dimension while the rest of our analysis focuses on the cases when n = 2 and n = 3 in two dimensions.

math.PR↗
arXiv2013

Stochastic spatial model of producer-consumer systems on the lattice

The objective of this paper is to give a rigorous analysis of a stochastic spatial model of producer-consumer systems that has been recently introduced by Kang and the author to understand the role of space in ecological communities in which individuals compete for resources. Each point of the square lattice is occupied by an individual which is characterized by one of two possible types, and updates its type in continuous time at rate one. Each individual being thought of as a producer and consumer of resources, the new type at each update is chosen at random from a certain interaction neighborhood according to probabilities proportional to the ability of the neighbors to consume the resource produced by the individual to be updated. In addition to giving a complete qualitative picture of the phase diagram of the spatial model, our results indicate that the nonspatial deterministic mean-field approximation of the stochastic process fails to describe the behavior of the system in the presence of local interactions. In particular, we prove that, in the parameter region where the nonspatial model displays bistability, there is a dominant type that wins regardless of its initial density in the spatial model, and that the inclusion of space also translates into a significant reduction of the parameter region where both types coexist.

math.PR↗
arXiv2013

The naming game in language dynamics revisited

This article studies a biased version of the naming game in which players located on a connected graph interact through successive conversations to bootstrap a common name for a given object. Initially, all the players use the same word B except for one bilingual individual who also uses word A. Both words are attributed a fitness, which measures how often players speak depending on the words they use and how often each word is pronounced by bilingual individuals. The limiting behavior depends on a single parameter: phi = the ratio of the fitness of word A to the fitness of word B. The main objective is to determine whether word A can invade the system and become the new linguistic convention. In the mean-field approximation, invasion of word A is successful if and only if phi > 3, a result that we also prove for the process on complete graphs relying on the optimal stopping theorem for supermartingales and random walk estimates. In contrast, for the process on the one-dimensional lattice, word A can invade the system whenever phi > 1.053 indicating that the probability of invasion and the critical value for phi strongly depend on the degree of the graph. The system on regular lattices in higher dimensions is also studied by comparing the process with percolation models.

math.PR↗
arXiv2013

The role of dispersal in interacting patches subject to an Allee effect

This article is concerned with a stochastic multi-patch model in which each local population is subject to a strong Allee effect. The model is obtained by using the framework of interacting particle systems to extend a stochastic two-patch model that has been recently introduced by Kang and the author. The main objective is to understand the effect of the geometry of the network of interactions, which represents potential migrations between patches, on the long-term behavior of the metapopulation. In the limit as the number of patches tends to infinity, there is a critical value for the Allee threshold below which the metapopulation expands and above which the metapopulation goes extinct. Spatial simulations on large regular graphs suggest that this critical value strongly depends on the initial distribution when the degree of the network is large whereas the critical value does not depend on the initial distribution when the degree is small. Looking at the system starting with a single occupied patch on the complete graph and on the ring, we prove analytical results that support this conjecture. From an ecological perspective, these results indicate that, upon arrival of an alien species subject to a strong Allee effect to a new area, though dispersal is necessary for its expansion, strong long range dispersal drives the population toward extinction.

math.PR↗
arXiv2012

The Axelrod model for the dissemination of culture revisited

This article is concerned with the Axelrod model, a stochastic process which similarly to the voter model includes social influence, but unlike the voter model also accounts for homophily. Each vertex of the network of interactions is characterized by a set of $F$ cultural features, each of which can assume $q$ states. Pairs of adjacent vertices interact at a rate proportional to the number of features they share, which results in the interacting pair having one more cultural feature in common. The Axelrod model has been extensively studied during the past ten years, based on numerical simulations and simple mean-field treatments, while there is a total lack of analytical results for the spatial model itself. Simulation results for the one-dimensional system led physicists to formulate the following conjectures. When the number of features $F$ and the number of states $q$ both equal two, or when the number of features exceeds the number of states, the system converges to a monocultural equilibrium in the sense that the number of cultural domains rescaled by the population size converges to zero as the population goes to infinity. In contrast, when the number of states exceeds the number of features, the system freezes in a highly fragmented configuration in which the ultimate number of cultural domains scales like the population size. In this article, we prove analytically for the one-dimensional system convergence to a monocultural equilibrium in terms of clustering when $F=q=2$, as well as fixation to a highly fragmented configuration when the number of states is sufficiently larger than the number of features. Our first result also implies clustering of the one-dimensional constrained voter model.

math.PR↗
arXiv2011

Consensus in the two-state Axelrod model

The Axelrod model is a spatial stochastic model for the dynamics of cultures which, similarly to the voter model, includes social influence, but differs from the latter by also accounting for another social factor called homophily, the tendency to interact more frequently with individuals who are more similar. Each individual is characterized by its opinions about a finite number of cultural features, each of which can assume the same finite number of states. Pairs of adjacent individuals interact at a rate equal to the fraction of features they have in common, thus modeling homophily, which results in the interacting pair having one more cultural feature in common, thus modeling social influence. It has been conjectured based on numerical simulations that the one-dimensional Axelrod model clusters when the number of features exceeds the number of states per feature. In this article, we prove this conjecture for the two-state model with an arbitrary number of features.

math.PR↗
arXiv2011

The role of space in the exploitation of resources

In order to understand the role of space in ecological communities where each species produces a certain type of resource and has varying abilities to exploit the resources produced by its own species and by the other species, we carry out a comparative study of an interacting particle system and its mean-field approximation. For a wide range of parameter values, we show both analytically and numerically that the spatial model results in predictions that significantly differ from its nonspatial counterpart, indicating that the use of the mean-field approach to describe the evolution of communities in which individuals only interact locally is invalid. In two-species communities, the disagreements between the models appear either when both species compete by producing resources that are more beneficial for their own species or when both species cooperate by producing resources that are more beneficial for the other species. In particular, while both species coexist if and only if they cooperate in the mean-field approximation, the inclusion of space in the form of local interactions may prevent coexistence even in cooperative communities. Introducing additional species, cooperation is no longer the only mechanism that promotes coexistence. We prove that, in three-species communities, coexistence results either from a global cooperative behavior, or from rock-paper-scissors type interactions, or from a mixture of these dynamics, which excludes in particular all cases in which two species compete.

math.DS↗
arXiv2010

Opinion dynamics with confidence threshold: an alternative to the Axelrod model

The voter model and the Axelrod model are two of the main stochastic processes that describe the spread of opinions on networks. The former includes social influence, the tendency of individuals to become more similar when they interact, while the latter also accounts for homophily, the tendency to interact more frequently with individuals which are more similar. The Axelrod model has been extensively studied during the past ten years based on numerical simulations. In contrast, we give rigorous analytical results for a generalization of the voter model that is closely related to the Axelrod model as it combines social influence and confidence threshold, which is modeled somewhat similarly to homophily. Each vertex of the network, represented by a finite connected graph, is characterized by an opinion and may interact with its adjacent vertices. Like the voter model, an interaction results in an agreement between both interacting vertices -- social influence -- but unlike the voter model, an interaction takes place if and only if the vertices' opinions are within a certain distance -- confidence threshold. In a deterministic static approach, we first give lower and upper bounds for the maximum number of opinions that can be supported by the network as a function of the confidence threshold and various characteristics of the graph. The number of opinions coexisting at equilibrium is then investigated in a probabilistic dynamic approach for the stochastic process starting from a random configuration ...

math.PR↗
arXiv2010

Two-scale multitype contact process: coexistence in spatially explicit metapopulations

It is known that the limiting behavior of the contact process strongly depends upon the geometry of the graph on which particles evolve: while the contact process on the regular lattice exhibits only two phases, the process on homogeneous trees exhibits an intermediate phase of weak survival. Similarly, we prove that the geometry of the graph can drastically affect the limiting behavior of multitype versions of the contact process. Namely, while it is strongly believed (and partly proved) that the coexistence region of the multitype contact process on the regular lattice reduces to a subset of the phase diagram with Lebesgue measure zero, we prove that the coexistence region of the process on a graph including two levels of interaction has a positive Lebesgue measure. The relevance of this multiscale spatial stochastic process as a model of disease dynamics is also discussed.

math.PR↗
arXiv2010

Expansion or extinction: deterministic and stochastic two-patch models with Allee effects

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment, focusing on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global in-all-patch extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and its stochastic analog. The deterministic model has either two or four attractors. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either an in-all-patch expansion or an in-all-patch extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the in-all-patch extinction and the basin of attraction of the in-all-patch expansion. Above this threshold, while increasing the intensity of the dispersal, both deterministic and stochastic models predict a synchronization of the patches resulting in either a global expansion or a global extinction: for the deterministic model, two of the four attractors present when the dispersal is weak are lost, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee threshold as the global population size in the deterministic and the stochastic two-patch models evolves as dictated by the single-patch counterparts ...

math.CA↗
arXiv2003

Ecological Succession Model

We introduce a new interacting particle system intended to model an example of ecological succession involving two species: the bracken and the european beech. The objective is to exhibit phase transitions by proving that there exist three possible evolutions of the system to distinct ecological balances. More precisely, the whole population may die out, the european beech may conquer the bracken, and coexistence of both species may occur.

math.PR↗