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# The fundamentals of cultural adaptation: implications for human adaptation | Scientific Reports

In the follow we consider a cultural trait with two alternate variants, a and A, where A is ancestral and a represents a novel invention. We calculate probability of fixation for the novel initiation in two different adaptation scenarios : adaptation from de novo invention and adaptation from standing magnetic declination. adaptation from de novo initiation occurs where the version a arose in a single individual ( i.e. with frequency 1/ N ) after an environmental switch and therefore corresponds to a situation where a beneficial initiation is found once the environment has changed. adaptation from standing version occurs where the novel variant a was present in the population at some frequency at the time of the environmental careen. These represent to a position where where a cultural reaction to a fresh environment is found after the environment has changed and one where the population ’ s existing cultural repertoire already contains a reaction to the changed environmental conditions, respectively .

### Model set-up

The population is finite containing N individuals each possessing one variant of the cultural trait, either a or A. In each time step a raw person arises and adopts variant a or A, before replacing another randomly selected individual, who dies 13. initially, neither form provides an adaptive benefit and both are transmitted from a randomly chosen role-model to a newborn with a probability equal to its frequency. In other words, the variants evolve neutrally, through indifferent transmittance. At time \ ( T_0\ ) an environmental shift occurs. After this, random variable a provides an adaptive benefit f and A provides a benefit g, with \ ( farad > g\ ). immediately, the modern individual chooses a character model with a probability weighted by g and f and adopts its role-model ’ mho cultural version. therefore, the variants a and A now evolve through payoff-biased transmission 5 and f can be interpreted as the cultural infection advantage of variant a. In both transmittance regimes, biased and unbiased, the time development of the total of variants of type a deliver in the population at time t can be modelled as a Markov process \ ( \ { X_t : metric ton \ge 0\ } \ ) with values in the plant \ ( \ { 0,1, \ldots, N\ } \ ). The procedure \ ( N-X_t\ ) describes the development of trait A .

### The probability of fixation from de novo innovation

We calculate the probability that the novel discrepancy a will fix given that it arose after the environmental change at frequency 1/ N. The passage probabilities for the Markov process \ ( X_t\ ) in this casing are given by

\begin{aligned} p_{i,i-1}= & {} \frac{g(N-i)}{fi+g(N-i)} \frac{i}{N}=\beta _i\\ p_{i,i+1}= & {} \frac{fi}{fi+g(N-i)}\frac{N-i}{N}=\alpha _i\\ p_{i,i}= & {} 1-p_{i,i+1}-p_{i,i-1}=1-\alpha _i-\beta _i, \quad i=1,\ldots ,N-1 \end{aligned}

where \ ( p_ { one, \cdot } \ ) describes the probability that the absolute frequency of version a in the population changes from i to \ ( i-1, i\ ) or \ ( i+1\ ) in one time step. Further, f quantifies the profit of version a after the environmental change and g the benefit of variant A ( assumed to be 1 in the trace ). It can be shown ( see Supplementary Section S1 in the auxiliary material for a detailed derivation ) that the probability of obsession from a de novo initiation with adaptive benefit f is given by

\begin{aligned} \pi _\text {DN} = \pi _1=\frac{1}{1+\sum \nolimits _{l=1}^{N-1}\prod \nolimits _{k=1}^{l}\frac{\beta _k}{\alpha _k}}. \end{aligned}

( 1 )

### The probability of fixation from standing variation

future, we assume that the initiation of random variable a occurred some time before the environmental variety and indifferent transmission has caused it to reach frequency j / N, with \ ( j=1, \ldots, N-1\ ) at \ ( T_0\ ). We condition on the universe of a variant a that has not yet reached fixation in the population and consequently, the probability of fixation of a after the environmental shift will depend not merely on the profit of a but besides on the ask frequency of a at \ ( T_0\ ). To calculate the arrested development probability we first calculate the probability that variant a has frequency j / N under indifferent transmittance and multiply this by the probability of fixation from frequency j. In the shell of unbiased transmission, the conversion probabilities of the Markov march \ ( X_t\ ) are given by

\begin{aligned} p_{i,i-1}=p_{i,i+1}= & {} \frac{i(N-i)}{N^2}=a_i,\\ p_{i,i}= & {} \frac{i^2+(N-i)^2}{N^2}=1-2a_i,\quad i=1,\ldots ,N-1. \end{aligned}

The probability that random variable a has reached frequency j / N is given by \ ( \frac { t_ { 1j } } { t_1 } \ ) where \ ( t_ { 1j } \ ) denotes the mean time that the Markov process \ ( X_t\ ) with the initial condition \ ( X_0=1\ ) was in state j and \ ( t_1\ ) is the mean time that variant a exists before assimilation into either state 0 or N. It holds ( see Supplementary Section S2 in the supplementary substantial for a detail derivation ) that

\begin{aligned} t_{1j}=\frac{N}{j} \end{aligned}

( 2 ) and

\begin{aligned} t_{1}=N\left( 1+\sum \limits _{k=2}^{N-1}\frac{1}{k}\right) \end{aligned}

( 3 ) which leads to

\begin{aligned} \frac{t_{1j}}{t_1} = \frac{1}{j\left( 1+\sum \nolimits _{k=2}^{N-1}\frac{1}{k}\right) }. \end{aligned}

finally, we generalise the saying for the fixation probability ( 1 ) from a start frequency of 1 to a general begin frequency of j using the fact that

\begin{aligned} \pi _j=\pi _1+\pi _1\sum _{l=1}^{j-1}\prod \limits _{k=1}^l\frac{\beta _k}{\alpha _k} \end{aligned}

and obtain

\begin{aligned} \pi _j=\frac{1+\sum _{l=1}^{j-1}\prod _{k=1}^{l}\frac{\beta _k}{\alpha _k}}{1+\sum _{l=1}^{N-1}\prod _{k=1}^{l}\frac{\beta _k}{\alpha _k}}. \end{aligned}

So the probability of a fixation from standing cultural variation, at the clock of an environmental transfer, i.e. from a form with frequency j / N at \ ( T_0\ ) and an adaptive profit f, is given by

\begin{aligned} \pi _\text {SV}=\sum _{j=1}^{N-1} \frac{t_{1j}}{t_{1}}\cdot \pi _j. \end{aligned}

( 4 ) Summarising, the probabilities \ ( \pi _\text { DN } \ ) ( 1 ) and \ ( \pi _\text { SV } \ ) ( 4 ) carry how likely trait a with profit f goes to fixation when it is a de novo invention or part of the existing cultural repertory of the population, respectively. human body 1 illustrates those probabilities for versatile values of f. The obsession probability is lowest if the adaptive trait is a de novo initiation, i.e. invented at \ ( T_0\ ) with frequency 1/ N for all values of f ( compare bolshevik line for de novo invention and black line for standing variation ). This is an intuitive line up as standing variation can result in situations where the frequency of the adaptive random variable a is larger than 1/ N at the meter of the environmental transfer what in turn leads to a higher fixation probability. Before we discuss some implications of these results for the theory of cultural adaptation, we consider the influence of transmission processes other than indifferent transmission on the fixation probability .

### The probability of fixation from standing variation under alternative transmission mechanisms

An important remainder between genetic and cultural development is the large number of different ways in which information can be passed on from one generation to the adjacent in a cultural context 14. cultural transmission processes affect how cultural traits are maintained or lost in a population 4. As a resultant role, it is potential that the probability of a embroil to arrested development might depend on the cultural transmission processes on which a population relies before an environmental deepen. To quantify the effects of alternative transmittance processes, we need to generalise Eq. ( 4 ) to allow for the general transition probabilities \ ( p_ { i, i-1 } =\beta _i, p_ { iodine, i+1 } =\alpha _i, \text { and } p_ { one, one } =1-\alpha _i-\beta _i\ ) We merely consider transmission processes whose temporal dynamic is markovian. In doing thus ( see Supplementary Section S3 in the supplementary material for a detailed derivation ) we obtain for the mean time to absorption, \ ( t_1\ ), and average time spent at a given frequency, \ ( t_ { 1j } \ )

\begin{aligned} t_{1j}=\frac{\frac{(N-j)}{\alpha _j}\prod \nolimits _{k=j+1}^{N -1}\frac{\beta _k}{\alpha _k}}{N\prod \nolimits _{l=1}^{N-1}\frac{\beta _l}{\alpha _l}}. \end{aligned}

( 5 ) and

\begin{aligned} t_{1}=\sum _{j=1}^{N-1}t_{1,j}. \end{aligned}

( 6 ) Substituting these expressions in Eq. ( 4 ) provides us with the obsession probability from standing variation assuming an arbitrary cultural transmission process defined by the transition probabilities \ ( \alpha _i\ ) and \ ( \beta _i\ ). In other words, we can derive the fixation probabilities for any cultural transmission work for which we can formulate the transition probabilities \ ( \alpha _i\ ) and \ ( \beta _i\ ) of the equate Markov process. We note that the population still applies payoff-biased infection after the environmental fault. To illustrate the likely effect of cultural transmission processes on the probability of a end run to fixation, we assume that transmittance before \ ( T_0\ ) is governed by a frequency-dependent bias, i.e. the leaning to disproportionately copy either common variants ( ossification ) or rare variants ( anti-conformity ) 5. In this case the transition probabilities are given by

\begin{aligned} p_{i,i+1}= & {} \frac{(i/N)^{(1+\theta )}}{(i/N)^{(1+\theta )}+(1-(i/N))^{(1+\theta )}}\frac{N-i}{N}=\alpha _i,\nonumber \\ p_{i,i-1}= & {} \frac{((N-i)/N)^{(1+\theta )}}{(i/N)^{(1+\theta )}+(1-(i/N))^{(1+\theta )}}\frac{i}{N}=\beta _i,\nonumber \\ p_{i,i}= & {} 1-\beta _i-\alpha _i. \end{aligned}

( 7 ) where \ ( \theta > 0\ ) models conformity and \ ( \theta < 0\ ) anti-conformity. Calculating the arrested development probability ( 4 ) using Eqs. ( 5 ) – ( 7 ) allows us to compare the probability of a cultural sweep under different transmittance processes anterior to \ ( T_0\ ). calculate 1 shows that, compared to unbiased transmittance ( see black solid production line ), conformity ( see short dashed line ) and anti-conformity ( see farseeing smash line ) picture higher arrested development probabilities for all values of f. This is because conformity reduces the probability that variant a has high frequency at \ ( T_0\ ), while anti-conformity increases the probability that a version is maintained at an intercede frequency. This is shown in Fig. 2, which shows the frequency distribution of a discrepancy for the three infection processes considered above ) .Figure 1 ( A ) The probability of a cultural selective end run from standing magnetic declination generated by unbiased transmittance ( black firm line ), conformity with \ ( \theta =0.5\ ) ( dashed-dotted production line ), anti-conformity with \ ( \theta =-0.5\ ) ( smash line ) or from de novo invention ( red line ) after an environmental variety. Full size personaFigure 2 probability that variant a has frequency j shown on the x-axis at \ ( T_0\ ) under ( A ) unbiased transmittance, ( B ) ossification with \ ( \theta =0.05\ ), and ( C ) anti-conformity with \ ( \theta =-0.05\ ). Full size persona As a side note, knowing the proportion \ ( t_ { 1j } /t_1\ ) allows us to derive a kind of ‘ trait frequency spectrum ’ for an infinite sites Moran model 15 under the transmission process defined by \ ( \alpha _i\ ) and \ ( \beta _i\ ). The issue of variants with frequency j in the population, denoted by \ ( S_ { N, j } \ ), is given by

\begin{aligned} S_{N,j}=\frac{t_{1j}}{t_1}S_N=t_{1j}\mu \end{aligned}

where \ ( S_N\ ) represents the average number of cultural variants expected to be present in the population at some timestep t and \ ( \mu\ ) the per capita initiation rate. ( for more detail witness supplementary Section S4 in the auxiliary material ) .

### Implications for the theory of cultural adaptation

The results therefore army for the liberation of rwanda have shown that if the standing cultural pas seul in a population contains a variant that becomes adaptive after an environmental change, then the probability of a sweep to obsession is higher compared to a position where an adaptive variant with the like level of benefit is invented after an environmental shift. This result is intuitive—standing variation is probably to produce variants with frequencies larger than 1/ N at \ ( T_0\ ) and these variants are at an advantage compared with those at lower frequency. These results suggest it is plausible that under some circumstances populations need not, and indeed should not, trust on inventing novel traits in novel environmental conditions if they possess adaptive stand variation. naturally this raises far questions such as ‘ under what circumstances do populations possess adaptive standing cultural version, and what mechanisms produce and maintain it ? ’. In other words, exploring the mechanisms that can generate standing mutant containing an adaptive discrepancy after an arbitrary environmental variety is of great interest. An extensive analysis of these questions might require an n variant exemplary to allow for the collection of cultural diverseness and this is a discipline of future research. In the next incision, as above, we explore in the two version model a elementary mechanism capable of generating and maintaining standing variation : prevision. We note that there are a numeral of candidate mechanisms capable of maintaining cultural variation such as frequent environmental changes, accurate copying of huge bodies of cultural cognition, relatively accurate foresight, or high initiation rates. here, we investigate just one childlike mechanism .

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