Random Replicators with High-Order Interactions

Oliveira, Viviane M. de; Fontanari, José F.

doi:10.1103/physrevlett.85.4984

Cited by 30 publications

(51 citation statements)

References 13 publications

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“…that interaction between species is pairwise and characterized by the matrix elements w ij . Generalization to multi-species interaction is possible [39,40], and will be discussed below. The matrix elements {w ij , w ji } (for any pair i < j) are chosen from a Gaussian ensemble.…”

Section: Modelmentioning

confidence: 99%

Rank abundance relations in evolutionary dynamics of random replicators

2008

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We present a non-equilibrium statistical mechanics description of rank abundance relations (RAR) in random community models of ecology. Specifically, we study a multi-species replicator system with quenched random interaction matrices. We here consider symmetric interactions as well as asymmetric and anti-symmetric 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 co-operation pressures as well as effects of finite system size and of higher-order interactions.

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Section: Modelmentioning

confidence: 99%

Rank abundance relations in evolutionary dynamics of random replicators

2008

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“…These describe the evolution of selfreproducing interacting species within a given framework of limited resources and have found wide applications in a variety of fields including game theory, socio-biology, prebiotic evolution and optimization theory [4,5,6]. While the first replicator system with quenched random couplings was introduced by Diederich and Opper in [7,8], most subsequent studies in the statistical physics community seem to be based on replica theory or on computer simulations [10,11,12,13,14,15,16,17,18]. Thus most of the existing analytic work on RRM is restricted to the case of symmetric couplings, in which a Lyapunov function can be found, so that replica theory is applicable.…”

Section: Introductionmentioning

confidence: 99%

Random replicators with asymmetric couplings

Galla¹

2006

J. Phys. A: Math. Gen.

103

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Abstract. The dynamics of random replicators is studied using generating functional techniques. While replica analyses are limited to models with symmetric couplings, dynamical approaches as presented here allow specifically to address cases with asymmetric interactions. We first discuss in detail how dynamical theories can be formulated for general replicator models in terms of an effective single-species process, and how persistent order parameters of the ergodic stationary states can be extracted from this process. We then detail how different types of dynamical phase transitions can be identified and related to each other. As an application of the general theory we address replicator models with Gaussian couplings of arbitrary symmetry between pairs and triples of species, respectively. Numerical simulations verify our theory, and also indicate regimes in which only a finite number of species survives, even when the thermodynamic limit is considered.

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“…The randomness of the interspecies couplings here reflects an amount of uncertainty about the structure of interactions in real ecosystems. The initial RRM and various extensions have subsequently been studied in a series of papers [8,9,10,11,12,13,14,15,16,17] and they have been found to exhibit intriguing features, both from the biological point of view as well as from the perspective of statistical mechanics. In particular it has been realised that the model exhibits phase behaviour with interesting ergodic and nonergodic phases, different types of phase transitions as well as replica-symmetry breaking.…”

Section: Introductionmentioning

confidence: 99%

“…The variations of RRMs mentioned above then differ in the details of the statistics from which the couplings are drawn. In the original model of Diederich and Opper the interactions between the species were assumed to be pairwise and drawn from a Gaussian distribution, [9] discusses the case of higher-order Gaussian interactions and [11], [12] and [13] are concerned with models in which the couplings are of a separable Hebbian structure, reminiscent of those used extensively in the context of neural networks [18,19].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of random replicators with Hebbian interactions

Galla¹

2005

J. Stat. Mech.

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Abstract. A system of replicators with Hebbian random couplings is studied using dynamical methods. The self-reproducing species are here characterized by a set of binary traits and interact based on complementarity. In the case of an extensive number of traits we use path-integral techniques to demonstrate how the coupled dynamics of the system can be formulated in terms of an effective single-species process in the thermodynamic limit, and how persistent order parameters characterizing the stationary states may be computed from this process in agreement with existing replica studies of the statics. Numerical simulations confirm these results. The analysis of the dynamics allows an interpretation of two different types of phase transitions of the model in terms of memory onset at finite and diverging integrated response, respectively. We extend the analysis to the case of diluted couplings of an arbitrary symmetry, where replica theory is not applicable. Finally the dynamics and in particular the approach to the stationary state of the model with a finite number of traits is addressed.

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Random Replicators with High-Order Interactions

Cited by 30 publications

References 13 publications

Rank abundance relations in evolutionary dynamics of random replicators

Rank abundance relations in evolutionary dynamics of random replicators

Random replicators with asymmetric couplings

Dynamics of random replicators with Hebbian interactions

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