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[EQUATION] where we have introduced [MATH] . This scaling law appears in models of anomalous diffusion by low-frequency turbulence PAppl PAppl+ Zim01 PRE01 PRE09 . Special cases of Eq. ( ) include the well-known Bohm scaling Bohm , characterized by [MATH] , as well as the anomalous so-called “percolation” scaling ( [MA... |
[EQUATION] where [MATH] is a scaling function, which interpolates between the initial-time power-law and flat asymptotic ( [MATH] ) behavior: [MATH] The form in Eq. ( ) is similar to that considered by Gefen et al. |
Gefen for anomalous diffusion on percolation clusters (in their model, [MATH] ), and earlier by Straley Straley Power-law power spectral density |
By applying the Kramers-Kronig relations [MATH] and [EQUATION] it is found that [MATH] , with [MATH] . A Fourier transformed Eq. ( ) reads [MATH] . One can see that the power spectral density, [MATH] , of the system response to a white-noise perturbation, [MATH] , will be proportional to [MATH] . The end result reads: |
[EQUATION] where [MATH] . The conclusion is that the power spectral density in the DPRW model is given by an inverse power-law distribution, with the [MATH] value depending on scaling properties of the ac conductivity response. |
Stretched-exponential relaxation and the distribution of relaxation times Next, we obtain the distribution of relaxation times self-consistently. For this, assume that the system is slightly supercritical, then consider a charge density perturbation, [MATH] , caused by the presence of either free charges or holes on th... |
[EQUATION] where, as usual, [MATH] for [MATH] . The density of relaxation currents is defined as the time derivative of [MATH] , i.e., |
[EQUATION] The continuity implies that [EQUATION] Taking [MATH] under the integral sign, then eliminating [MATH] by means of Maxwell’s equation [MATH] , we find, with the self-consistent charge density, |
[EQUATION] In writing Eqs. ( 14 ) and ( 15 ) we have also assumed that [MATH] . We now integrate in Eq. ( 15 ) to find [EQUATION] |
Here, the function [MATH] is an arbitrary function of the position vector [MATH] , which appears in the derivation as the constant of integration over time. Under the conditions [MATH] for [MATH] and [MATH] for [MATH] for all [MATH] , Eq. ( 16 ) reduces to |
[EQUATION] If we allow [MATH] , we find that for [MATH] the integral term on the left-hand-side goes to zero (as [MATH] ): [EQUATION] |
from which it is clear that [MATH] . We consider this last condition as the initial condition for the relaxation problem. Essentially the same condition holds in the limit [MATH] , provided that [MATH] is taken first. A Fourier transformed Eq. ( 17 ) reads |
[EQUATION] where [MATH] is position vector in reciprocal space, and [MATH] is the Fourier image of [MATH] . Writing the susceptibility as [MATH] with [MATH] a time constant it is found that |
[EQUATION] The quantity [MATH] has the sense of lifetime of a perturbation with wavelength [MATH] . We expect that [MATH] at criticality, where [MATH] is a scaling exponent. A derivation of this scaling relation will be given shortly. Separating the variables, we write [MATH] , with |
[EQUATION] which we consider as the relaxation function in the frequency domain. On inversion to the time domain, Eq. ( 21 ) generates the Mittag-Leffler function, [MATH] , which has series expansion Mittag-Leffler Klafter |
[EQUATION] Thus, [MATH] . One sees that the relaxation to SOC of a supercritical state is described by the Mittag-Leffler function [MATH] , and not by a simple exponential function as for standard relaxation. We note in passing that the Mittag-Leffler function is the natural generalization of the exponential function. ... |
[EQUATION] which is often found empirically in various amorphous materials as for instance in many polymers and glass-like materials near the glass transition temperature (for reviews see Refs. Phillips and Kaatz , and references therein). The KWW relaxation function can conveniently be considered Montroll as a weighte... |
[EQUATION] The weighting function [MATH] is given by Eqs. (51d) and (55) of Ref. Montroll where one replaces the exponent [MATH] with [MATH] , the time constant [MATH] with [MATH] , and the variable [MATH] with [MATH] . In our notations: |
[EQUATION] where [MATH] is the Lévy distribution function with skewness [MATH] (e.g., Ref. Wolf ). Assuming a long-wavelength perturbation (i.e., the parameter [MATH] being much longer than the microscopic lattice distance: [MATH] ), and setting [MATH] , we can further approximate the Lévy distribution [MATH] by the Pa... |
[EQUATION] These distributions were earlier conjectured for SOC Tang . Our conclusion so far is that the relaxations are multi-scale, in accordance with Eq. ( 24 ), and their durations are power-law distributed. The distribution is heavy-tailed in the sense that [MATH] for [MATH] |
Consistency check In a basic theory of dielectric relaxation one writes the frequency-dependent complex dielectric parameter as Montroll Constant |
[EQUATION] where [MATH] is the relaxation function that describes the decay of polarization after the polarizing electric field has been stepped down or removed instantaneously. In the DPRW model, a step-down type electric field occurs as a consequence of re-injection of the free charges to the infinite cluster. The en... |
[EQUATION] where [MATH] is dimensionless frequency, and [MATH] and [MATH] are the Lèvy definite integrals: [EQUATION] [EQUATION] |
In the parameter range of multi-scale relaxation response, [MATH] [MATH] , the following series expansions of the Lèvy integrals hold Montroll |
[EQUATION] [EQUATION] From Eqs. ( 31 ) and ( 32 ) one can see that the expansion of [MATH] starts from a term which is proportional to [MATH] , and so does the expansion of [MATH] . Hence, up to higher order terms, [MATH] . Given this, one applies the Kramers-Kronig relations [MATH] and |
[EQUATION] to find the scaling of the ac conduction coefficient to be [MATH] . By comparing this with the above expression [MATH] one reiterates that [MATH] consistently with the distribution of durations of relaxation events, Eq. ( 26 ). |
Fractional relaxation and diffusion equations As was shown by Glöckle and Nonnenmacher Nonn , the Mittag-Leffler function, Eq. ( 22 ), is the solution of the fractional relaxation equation |
[EQUATION] where [EQUATION] is a fractional time the so-called Riemann-Liouville derivative Klafter Podlubny . Partial cases of this derivative are the unity operator for [MATH] and [MATH] for [MATH] . It is noticed, following Ref. Sokolov , that the Mittag-Leffler function [MATH] describes the relaxation toward equili... |
[EQUATION] where [MATH] is the probability density of finding a particle (random walker) at time [MATH] at point [MATH] , and the Laplacian operator stands for the local (nearest-neighbor) character of the lattice interactions. |
Derivation of the fractional diffusion equation It is instructive to obtain the fractional diffusion equation, Eq. ( 36 ), directly from the DPRW relaxation model. For this, let us introduce the electrostatic potential, [MATH] , corresponding to the electric field inhomogeneity, [MATH] . Upon substituted into Eq. ( 14 ... |
[EQUATION] In the vicinity of self-organized critical state, we can represent the total charge density, [MATH] , as a sum of “unperturbed” or background density, [MATH] , and a perturbation, [MATH] , describing the deviation from criticality: [MATH] . To obtain the dependence of [MATH] , we use the effective-medium app... |
[EQUATION] Self-consistently, one requires that, on the average, the embedding in the effective medium has the same overall property as the effective medium itself Dyre Effective . In writing Eq. ( 38 ) we took into account that the perturbation, [MATH] , is due to negatively charged particles (electrons and/or holes).... |
[EQUATION] The memory function, [MATH] , is obtained as Fourier inversion of [MATH] , yielding [MATH] . Under the conditions [MATH] for [MATH] and [MATH] for [MATH] for all [MATH] , the improper integration in Eq. ( 39 ) can be performed in the limits from 0 to [MATH] . Collecting all dimensional and numerical paramete... |
[EQUATION] which is an equivalent form of Eq. ( 36 ). The fractional diffusion equation, Eq. ( 40 ), can be thought of as deriving from the generalized “Fick’s law” |
[EQUATION] or [MATH] , which is readily deduced from Eq. ( 13 ) in the effective-medium approximation. Equation ( 41 ) can equivalently be obtained PRE09 from the general scaling law for anomalous diffusion on percolation systems. A derivation using the scheme of continuous time random walks (CTRW’s) can be found in Re... |
The occurrence of the fractional diffusion equation, Eq. ( 40 ), might be interpreted, with the aid of the proposed SOC model, in favor of considering SOC as one important case for fractional kinetics |
Nature . The concept of fractional kinetics enters different areas of research, such as turbulent transport in plasmas and fluids, particle dynamics in potential fields, quantum optics, and many others. This subject is summarized in comprehensive reviews Klafter Report Klafter2 . In many ways equations built on fractio... |
Dispersion-relation exponent, Hurst exponent, and the [MATH] -exponent In sandpile SOC models, one is interested in how the lifetime of an activation cluster scales with its size Zhang . In the DPRW model by activation cluster one means a connected cluster of activated sites. An occupied site is said “activated” if it ... |
Consider an isotropic activation cluster composed of the free particles. (The nature of the particles does not matter here [MATH] the hole case is just similar.) It is assumed for convenience, without loss of generality, that each site of the activation cluster contains only one particle. Thus, the number density of th... |
Tang , where one replaces [MATH] with [MATH] , and the fractal dimension [MATH] with the fractal dimension of the infinite percolation cluster, [MATH] . The end result is [MATH] . Note that the [MATH] values in Refs. Tang and Zhang differ by 1. |
Occurrence frequency energy distribution and the [MATH] -exponent It is convenient to think of the activation clusters as containing a certain amount of “energy” which is released when the comprising free particles dissipate to the boundaries. Using here that the electric charge of the free particles is a conserved qua... |
[EQUATION] where [MATH] is a power-law slope and [MATH] in the “hyperuniversal” fractal dimension. We interpret this distribution as the familiar from applications Char Hudson occurrence frequency energy distribution. Indeed the distribution in Eq. ( 42 ) might find its significance in the statistics of solar flares an... |
Values of the critical exponents Using known estimates Stauffer Isi Naka of the percolation indices [MATH] [MATH] , and [MATH] we could evaluate the critical exponents of the DPRW model in all ambient dimensions [MATH] . The results of this evaluation, summarized in Table 1, are in good agreement with the reported nume... |
In many ways, the DPRW approach to SOC offers a simple yet relevant lattice model for dielectric relaxation phenomena in systems with spatial disorder. One by-product of this approach is the case for stretched-exponential the KWW relaxation function, Eq. ( 23 ), which is often found empirically in various amorphous mat... |
More so, the DPRW model gives a Hurst exponent (for [MATH] [MATH] ; for [MATH] [MATH] ) consistently with the reported narrow range of variation of [MATH] as observed in different magnetic confinement systems (Hurst exponent varying between [MATH] and [MATH] Carreras Pedrosa Carreras2 Carreras3 . In this connection, it... |
With respect to the occurrence frequency energy distribution, Eq. ( 42 ), the model predicts that [MATH] in one dimension (for [MATH] [MATH] ) and [MATH] in the mean-field limit (for [MATH] [MATH] [MATH] , and [MATH] ). These results are exact. Also, one finds, approximately, [MATH] for [MATH] and [MATH] for [MATH] (Ha... |
All in all, the exponent [MATH] agrees well with the reported slopes of the occurrence frequency energy distribution for solar flares (around [MATH] to [MATH] Hudson Crosby Uchida , demonstrating that the observed power-law distribution of flare energy release is well reproduced under the assumption that the solar coro... |
# Source: arxiv 1208.0482 # Title: The concurrent evolution of cooperation and the population structures that support it # Sections: all # Downloaded: 2026-03-03T01:59:19.599792+00:00 |
The Concurrent Evolution of Cooperation and the Population Structures that Support it Abstract The evolution of cooperation often depends upon population structure, yet nearly all models of cooperation implicitly assume that this structure remains static. This is a simplifying assumption, since most organisms possess g... |
group size, relatedness, kin selection, multi-level selection, linkage disequilibrium, Snowdrift game It is widely appreciated that population structure drives the evolution of social traits. Cooperative behaviours, that benefit other individuals at some cost to the actor, can evolve if the population structure is such... |
Population structure is the product not only of environmental factors, but also of individual behaviours. Many of these behaviours that affect population structure have a genetic basis, and so are themselves subject to natural selection. For example, the evolution of individual traits that affect group size (Rodman 198... |
We thus consider a population structure that initially approaches freely-mixed conditions, where individuals have fitness-affecting interactions with many others, and then show how greater interaction structure can evolve. Specifically, this illustrates how evolution of an individual group size preference can increase ... |
Our approach is in contrast to recent work by Avilés ( 2002 , which shows how solitaires can evolve to live in groups (i.e., how a starting group size of 1 can evolve upwards). This fundamental difference is explained by the type of cooperative act that the respective models seek to explain. Specifically, Avilés consid... |
By contrast, we focus on indirect benefits that arise through increased kin or group selection for greater (individually-costly) cooperation. That is, we show how the benefits of cooperation can drive the evolution of structures that increase relatedness, and hence create the conditions for effective kin selection. |
The kind of social interactions that we consider can be readily modelled using the Prisoner’s Dilemma and Snowdrift games. Such evolutionary game theoretic models are commonly used to conceptualise problems of cooperation across taxa (Maynard Smith 1982 , including in the literature on cooperation in animals, humans, a... |
The benefits of cooperation can drive the evolution of population structure Peck and Feldman ( 1988 and Breden and Wade ( 1991 argue that cooperation can drive the evolution of population structure, for the specific case of evolution of aspects of the mating system. Here, we develop a general argument that applies in p... |
Consider two possible population structures, such as two different group sizes or two different mating systems. Suppose that structure [MATH] causes selection to favour more cooperative behaviour amongst its inhabitants, while structure [MATH] causes selection for more selfish behaviour. For example, structure [MATH] m... |
Moreover, the component of selection on population structure which derives from selection on social behaviour must favour the creation of structures that support cooperation, rather than selfishness (Powers 2010 |
The above argument makes two critical but logical assumptions. The first of these is that individuals with structural allele [MATH] find themselves living in structure [MATH] , whereas individuals with allele [MATH] find themselves living in structure [MATH] . Essentially, what matters is that a structural differential... |
In the model presented below, we consider the introduction of new population-structuring alleles by small mutations from existing ones. This model serves to illustrate the logical argument presented above. It also serves to elucidate what the assumptions of the argument mean for the specific case of group size evolutio... |
The concurrent evolution of initial group size preference and public goods production To illustrate the above argument, we consider the concurrent evolution of the number of individuals that found a group (which we hereafter refer to as founding size or simply group size) with public goods production. Public goods are ... |
However, the “tragedy” can potentially be averted in a group-structured population. In group-structured populations, individual selection on social behaviour (kin selection) can be partitioned into two components (Price 1972 ; Hamilton 1975 ; Wilson 1975 , as follows. First, within each social group cooperators may dec... |
The amount of cooperation that evolves in a particular case depends on the proportion of the total genetic variance, at the locus for public goods production, that is between groups (Wilson 1975 ; Hamilton 1975 (or equivalently, the genetic relatedness of group members (Queller 1992 ; Lehmann et al. 2007 ; see Discussi... |
The model Our model is based on the classic “Haystack”, or aggregation and dispersal, model initially developed by Maynard Smith ( 1964 and later expanded into a general multi-level selection model (e.g., Wilson and Colwell 1981 Wilson 1987 and Fletcher and Zwick 2004 ). We model a population of [MATH] haploid asexuall... |
Let [MATH] be a counter for the generations within groups in a single aggregation and dispersal cycle; it thus ranges from 0 to [MATH] , and is reset with every new cycle. Then, let [MATH] be the size of a group at generation [MATH] . We denote the founding size of a new group at the beginning of a cycle, [MATH] , by [... |
[EQUATION] where [MATH] is the total count of groups of founding size [MATH] , calculated as described above, [MATH] is the number of cooperators with size preference allele [MATH] , and [MATH] is the total number of individuals (cooperative and selfish) with size preference allele [MATH] . The term in square brackets ... |
The function [MATH] gives the distribution of individuals to groups at each group formation (aggregation) stage. After group formation, reproduction and selection occur within each group for [MATH] generations. In each generation the number of cooperators ( [MATH] ) and selfish individuals ( [MATH] ) in a group, and he... |
[EQUATION] where [MATH] , defined below, is the fitness payoff that a cooperator in a group of size [MATH] with [MATH] other cooperators receives from social interactions, [MATH] , also defined below, is the fitness payoff a selfish individual in the same group receives, and [MATH] is a baseline fitness in the absence ... |
Finally the number of cooperators in the global population with size preference allele [MATH] [MATH] , at the end of an aggregation and dispersal cycle, [MATH] , (i.e., after group formation by Equation , and [MATH] generations of reproduction and fitness-proportionate selection within groups given by [MATH] iterations... |
[EQUATION] This is the number of cooperators contributed to the global population by all possible groups of founding size [MATH] , multiplied by the expected count of that type of group (where the number of cooperators when the group is founded, [MATH] , varies from 0 to [MATH] , and [MATH] is the number of cooperators... |
[EQUATION] Our model consists of repeated iterations of equations and (which depend on quantities from equations ), corresponding to repeated aggregation and dispersal cycles. |
Within-group payoff functions We model the fitness payoffs that cooperative producers and selfish non-producers of the public good within a group receive using the [MATH] -player Prisoner’s Dilemma and Snowdrift games (Doebeli and Hauert 2005 , where [MATH] is the group size. The difference between the [MATH] -player P... |
This describes a negative frequency-dependent selection scenario, leading to a polymorphism of cooperative and selfish behaviours within a social group. |
The payoff matrix for the 2-player version of both the Prisoner’s Dilemma and Snowdrift games is shown in Table . Under this payoff structure, cooperators provide a benefit [MATH] to themselves and their social partner, at a cost to themselves of [MATH] . If their partner also cooperates, however, then the cost is shar... |
[EQUATION] In both the Snowdrift and Prisoner’s Dilemma parameterisations of these functions, individual selection within a group leads to an outcome that is suboptimal for the group, in terms of mean fitness of the group members. Specifically, although in both scenarios 100% cooperate provides the highest mean fitness... |
If these payoff functions were iterated within a single group until an equilibrium was reached, then the proportion of cooperators within that one group would be [MATH] |
(Doebeli and Hauert 2005 . This corresponds to the selfish allele at fixation within the group under the Prisoner’s Dilemma parameterisation, and a stable polymorphism of cooperative and selfish alleles under the Snowdrift game parameterisation. Thus, selection on social behaviour is directional in the first case, and ... |
(Wilson 1975 1980 Our use of game theoretic payoff functions does have the limitation that it assumes discrete cooperative and selfish phenotypes, rather than the more realistic case of the social phenotype representing a continuous degree of investment in the public good, or the probability that an individual contribu... |
Numerical analysis We examine the generation of linkage disequilibrium between socio-behavioural and population-structuring alleles, and the consequent selection for small founding group sizes that support cooperation, below. Closed form analysis for the generation of linkage disequilibrium in a social setting is known... |
Our approach is to consider a population initially fixed for a large group size preference allele, representing low between-group variance and hence little selection for cooperation, and then examine conditions under which mutant smaller group size alleles will be selected and increase between-group variance and select... |
Parameter settings The parameter settings stated in Table are used for the simulations presented below. We set [MATH] , with [MATH] to yield Prisoner’s Dilemma interactions, and [MATH] to produce a Snowdrift scenario (this is close to the qualitative threshold between the two types of selection on social behaviour that... |
Directional selection on social behaviour represented by the Prisoner’s Dilemma within groups We have investigated whether an individual adaptive gradient towards cooperative groups with high relatedness exists, and whether it can be followed when new group size preference alleles arise at mutation frequency. We first ... |
To verify that selection for a smaller size allele would not occur without the generation of positive linkage disequilibrium with cooperation, we removed the possibility for sustained linkage disequilibrium. We did this by forming groups based on the size preference allele as before, but by drawing the behavioural alle... |
Closer inspection of the results where linkage disequilibrium develops shows that the selection pressure favouring small groups is punctuated, not gradual. The plot of size allele frequencies over time in Figure shows that in the initial stages there is no significant component of selection pressure on population struc... |
Why, then, is there no significant selection pressure on founding group size in the initial stages? If a mutant smaller size preference is to be selectively advantageous, it must be the case that groups of that size have members which experience a greater frequency of cooperation than those of the current size. This fo... |
(Powers et al. 2008 . The solid line in Figure explores this in the case where social interactions follow an [MATH] -player Prisoner’s Dilemma, by calculating the frequency of cooperation in the global population over a range of non-evolving founding group sizes (taken over the last 1000 cycles, and averaged over 100 r... |
Negative frequency-dependent selection on social behaviour represented by the Snowdrift game within groups As previously discussed directional selection against cooperation, as modelled by the Prisoner’s Dilemma, represents a worst-case scenario. Rather, the nature of cooperation may be such that selection is negative-... |
To confirm this, we again considered whether a mutation from group size preference [MATH] to [MATH] would increase the amount of cooperation its bearers experienced. The dashed line in Figure shows that if all groups are of (founding) size [MATH] , then decreasing group size by 1 always yields some increase in cooperat... |
The results presented so far are from a model in which there is no direct selection on the group size preference allele. This has allowed us to show that cooperation can exert indirect selection pressure on group size preference, driving its evolution. However, we might expect there to be other direct components of sel... |
Analysis with an opposing component of selection on population structure due to an Allee effect We introduce here a direct component of selection into the model that favours larger groups, representing a (weak) Allee effect (Allee 1938 ; Odum and Allee 1954 ; Avilés 1999 . A larger founding group size may be favoured b... |
This component of selection is represented by the following, positive density-dependent, function of group size, [MATH] [EQUATION] |
[MATH] is a sigmoidal function of group size, with gradient [MATH] (which determines how quickly the benefit tails off as the group grows), and [MATH] a parameter which determines the maximum benefit. A sigmoidal function of group size is used to model the fact that, above a certain size, the advantages of number becom... |
[EQUATION] To obtain the numerical results presented below, we set [MATH] and [MATH] . In the absence of any possibility for cooperation, a founding group size of 20 or greater would be favoured using these parameter settings for the sigmoidal function, since this is the group size for which the gradient reaches zero. ... |
The results in Figure show how the evolution of founding group size is affected by the incorporation of an Allee effect. Founding group size preference evolves downwards, and between-group variance and hence cooperation increase, under Snowdrift but not Prisoner’s Dilemma types of interaction (figures and ). This is be... |
Interestingly, Figure suggests that the mean value of the size preference allele starts to increase again under Snowdrift interactions once the cooperative allele has reached a high frequency. This is because at the end of a run linkage disequilibrium exists between the size preference and behavioural alleles, such tha... |
Sensitivity analysis We have illustrated how a trait affecting the population structure of its bearers can evolve concurrently with social behaviour, and how a sharp qualitative distinction arises between Prisoner’s Dilemma and Snowdrift style social interactions. In particular, the distinction is over whether a small ... |
Discussion At the beginning of this article, we argued that cooperation could drive the evolution of population structure. The model presented above provides a simple illustration of this argument, for the particular population-structuring trait of founding group size preference. The model illustrates how a genetic pre... |
This argument hinges on the development of linkage disequilibrium between small size preference and cooperative alleles. For this to occur by selection, smaller groups must select for a greater degree of cooperation, since the only source of differential fitness in the model (with no Allee effect) is that caused by exp... |
There is a growing interest in whether selection for cooperation is directional or negative frequency-dependent in a range of biological scenarios (Doebeli and Hauert 2005 Gore et al. ( 2009 have recently verified empirically that the public goods scenario of extra-cellular enzyme production in yeast does indeed follow... |
Thus, weakly altruistic traits subject to negative frequency-dependent selection (the Snowdrift game) could, by inducing selection on population structure, scaffold the subsequent evolution of strong altruism (the Prisoner’s Dilemma). Through this mechanism, such traits may play an important role in the origin of high ... |
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