Dynamics of random replicators with Hebbian interactions

Galla, Tobias

doi:10.1088/1742-5468/2005/11/p11005

Cited by 13 publications

(46 citation statements)

References 26 publications

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“…we consider only u = w = 0. We furthermore follow the philosophy of introducing dilution in the context of neural networks [29] and in random replicator models [12], and assume that only a fraction c ∈ (0, 1] of interaction links between species is present. If an interaction between species i and j is present, then we take it to be determined by their respective use of resources, following [9].…”

Section: Effects Of Dilutionmentioning

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: Effects Of Dilutionmentioning

confidence: 99%

“…. , N. We also note that we use Q The analysis of the dilute model is straightforward and can be performed along the lines of [29,30,12]. The effective process reads: (30) and with all other definitions as in the fully connected model.…”

Section: Effects Of Dilutionmentioning

confidence: 99%

Statistical mechanics and stability of a model eco-system

2007

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“…Firstly, the study of the present model extends the statistical mechanics analysis of existing replicator models [5]-[7], [11,12] and relates to studies of Minority Games [1, 2]. Complex phase behaviour and different patterns of ergodicity breaking and instabilities have been identified in such models, with similarities as well as differences between replicator-type models and other systems.…”

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confidence: 64%

“…to model parameters for which the assumed fixed point is independent of initial conditions. The following self-consistent equations of the persistent order parameters {Q, χ, ν} can then be derived along the lines of [6,11,12]:…”

Section: Effective Macroscopic Theory and Fixed Point Ansatzmentioning

confidence: 99%

“…In the case of an eco-system this is an obvious drawback, as competitive interaction of, for example, prey-predator pairs cannot appropriately be addressed. A direct study of the dynamical equations allows us to extend the analysis to cases of asymmetric interaction matrices, and in particular to discuss the effects of anti-correlation on the behaviour of the system [11,12]. Asymmetric interaction comes in two ways in the present eco-system.…”

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I₂-Resistant TiO_x/Ag Collector Fabricated by Arc Plasma Deposition for Dye-Sensitized Solar Cells

Yoshida

Noma

Kashiwa

et al. 2008

jjap

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We study a model eco-system 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 eco-system. It is shown that resources can be exploited optimally only in the absence of co-operation pressure or direct interaction between species. This approach has been used to study, for example, the effects of self-interaction and memory in models of financial trading [1,2] or to examine how co-operation pressure and order of interactions impact on the stability and trajectories of replicator systems of evolutionary game theory [5]- [7]. In the context of population dynamics models with random interactions were first addressed by May in [8].In this paper we study a model of a simple food-web composed of species and resources, originally proposed in a more basic form in [9]. The level of the resource consumption by species and its relationship with the stability of the eco-system and the species richness is one of the main issues in ecology [10]. In [9], interaction between species is not through direct interaction (e.g. via prey-predator relations) but exclusively through the use and dependence on resources. If, for example, species A consumes a resource which B feeds upon as well, then this introduces a negative and symmetric interaction between A and B. The strength of the negative interaction between A and B is hence regulated by the overlap in their dependence on resources.Due to the symmetry of interactions the discussion of [9] focuses on a static analysis of this model eco-system. Here we choose a complementary dynamical approach, which allows us to address a broader class of interaction modes. Static studies necessarily rely on the existence of a Lyapunov function, extremized by the trajectories of the eco-system, and are hence limited to systems with symmetric interactions. In the case of an eco-system this is an obvious drawback, as competitive interaction of, for example, prey-predator pairs cannot appropriately be addressed. A direct study of the dynamical equations allows us to extend the analysis to cases of asymmetric interaction matrices, and in particular to discuss the effects of anti-correlation on the behaviour of the system [11,12]. Asymmetric interaction comes in two ways in the present eco-system. Firstly, we introduce direct interaction between species, in addition to the indirect interaction through the use and dependence on resources. Secondly, we study the effects of possibly asymmetric dilution of the network of interacting species.The aim of o...

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