Random replicators with asymmetric couplings

Galla, Tobias

doi:10.1088/0305-4470/39/15/001

Cited by 47 publications

(106 citation statements)

References 29 publications

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“…Simulations furthermore indicate that φ might actually vanish at small enough α and negative co-operation pressure indicating the possible existence of a phase in which only a sub-extensive number of species survives. Such behaviour has previously been reported for the case of higher-order interaction in [11]. Due to the limited relevance of negative co-operation pressure we have however not conducted a more detailed analysis of these observations, and can at this stage not fully confirm the existence of such a phase in this system of two-body interaction.…”

Section: No Direct Species Interactionmentioning

confidence: 67%

“…For general symmetry Γ no such Lyapunov function can be found, and the analysis needs to deal directly with the microscopic dynamics. The method of choice is here based on generating functionals, originally proposed in the context of random replicators in [6], and recently used in [11,12]. The analysis focuses on the dynamic partition function…”

Section: Effective Macroscopic Theory and Fixed Point Ansatzmentioning

confidence: 99%

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Statistical mechanics and stability of a model eco-system

2007

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Abstract. We study a model ecosystem by means of dynamical techniques from disordered systems theory. The model describes a set of species subject to competitive interactions through a background of resources, which they feed upon. Additionally direct competitive or co-operative interaction between species may occur through a random coupling matrix. We compute the order parameters of the system in a fixed point regime, and identify the onset of instability and compute the phase diagram. We focus on the effects of variability of resources, direct interaction between species, co-operation pressure and dilution on the stability and the diversity of the ecosystem. It is shown that resources can be exploited optimally only in absence of co-operation pressure or direct interaction between species.

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Section: No Direct Species Interactionmentioning

confidence: 67%

Section: Effective Macroscopic Theory and Fixed Point Ansatzmentioning

confidence: 99%

Statistical mechanics and stability of a model eco-system

2007

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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%

“…It is also applied to linear evolutionary dynamics model in [44] and can be adapted to the study of random Lotka-Volterra communities [45]. We will not detail the mathematical steps here, as they have been reported in depth in the literature [28,40]. In the thermodynamic limit the system is found to be described by an effective single-species process of the form [28] …”

Section: A Path Integral Analysismentioning

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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“…These are simply the relations of the arguments of the delta functions in eqs. (34,35). In this way, the cavity fields and biases are calculated self-consistently and the replica result is interpreted.…”

Section: Analysis Based On the Replica And Cavity Methodsmentioning

confidence: 99%

Relative species abundance of replicator dynamics with sparse interactions

2016

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Abstract. A theory of relative species abundance on sparsely-connected networks is presented by investigating the replicator dynamics with symmetric interactions. Sparseness of a network involves difficulty in analyzing the fixed points of the equation, and we avoid this problem by treating large self interaction u, which allows us to construct a perturbative expansion. Based on this perturbation, we find that the nature of the interactions is directly connected to the abundance distribution, and some characteristic behaviors, such as multiple peaks in the abundance distribution and all species coexistence at moderate values of u, are discovered in a wide class of the distribution of the interactions. The all species coexistence collapses at a critical value of u, u c , and this collapsing is regarded as a phase transition. To get more quantitative information, we also construct a non-perturbative theory on random graphs based on techniques of statistical mechanics. The result shows those characteristic behaviors are sustained well even for not large u. For even smaller values of u, extinct species start to appear and the abundance distribution becomes rounded and closer to a standard functional form. Another interesting finding is the non-monotonic behavior of diversity, which quantifies the number of coexisting species, when changing the ratio of mutualistic relations ∆. These results are examined by numerical simulations, and the multiple peaks in the abundance distribution are confirmed to be robust against a certain level of modifications of the problem. The numerical results also show that our theory is exact for the case without extinct species, but becomes less and less precise as the proportion of extinct species grows.

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Random replicators with asymmetric couplings

Cited by 47 publications

References 29 publications

Statistical mechanics and stability of a model eco-system

Statistical mechanics and stability of a model eco-system

Rank abundance relations in evolutionary dynamics of random replicators

Relative species abundance of replicator dynamics with sparse interactions

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