Package 'learnPopGen'

Illustrates the concept of the Central Limit Theorem

Description

The Central Limit Theorem tells us that when independent random variables are added together, the distribution of their sum tends towards a normal distribution, regardless of the shape of their individual distributions. This function attempts to illustrate this concept by allowing the user to visualize the sum of an arbitrary number of different independent random variables with different underlying distributions.

Usage

clt(nvar=1, nobs=1000, df=c("normal","uniform","exponential","binomial"),
		theta=NULL,	breaks="Sturges", show=c("sum","mean"))
	## S3 method for class 'clt'
print(x, ...)
	## S3 method for class 'clt'
plot(x, ...)

Arguments

nvar	number of random variables to sum (1 or more).
nobs	total number of observations (per random variable).
df	distribution functions of individual random variables to sum. These can be "normal", "uniform", "exponential", or "binomial".
theta	parameter of the distribution functions: variance in the case of df = "normal", maximum value in the case of "uniform" (the minimum value will be assumed to be 0), rate in the case of "exponential".
breaks	breaks (see hist).
show	whether to show the row-wise sum of the independent random variables (show="sum"), or their mean (show="mean").
x	object of class "clt" for print and plot methods.
...	optional arguments for print and plot methods.

Details

The central limit theorem (CLT) establishes that when independent random variables are added together their (normalized) sum will tend towards a normal distribution, regardless of the distribution of the original random variables. That is to say if we were to generate a set of nvar (say) independent, uniform, random variables, normalize each one to have the same variance, and then sum or average the variables by observation, this sum or average will tend towards a normal distribution as the number of random variables (nvar in this function) is increased.

Value

Creates a plot showing the observation-wise distribution of the sum or average of the independent random variables.

The distribution of the observation-wise sum or average and the underlying data are also returned invisibly to the user in the form of an object of class "clt". This object can in turn be printed or re-plotted using custom print and plot methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

phenotype.freq

Examples

clt(nvar=1,df="exponential")
	clt(nvar=10,df="exponential")
	object<-clt(nvar=40,df="exponential")
	print(object)
	plot(object)

---

Creates a (usually animated) simulation of gene coalescence within a population

Description

Coalescence or coalescent theory is a model for genetic drift within a population in which we envision gene copies merging or "coalescing" into ancestors in the past. This function generates a(n) (optionally animated) visualization of this process of coalescence within a population.

Usage

coalescent.plot(n=10, ngen=20, colors=NULL, ...)
	## S3 method for class 'coalescent.plot'
print(x, ...)
	## S3 method for class 'coalescent.plot'
plot(x, ...)

Arguments

n	number of haploid individuals or gene copies.
ngen	number of generations.
colors	colors to use for plotting individuals and lines. By default, the function tries to use a contrasting color scheme such that adjacent allele copies are dissimilar (to facilitate visualization of the coalescent process.)
x	object of class "coalescent.plot" for print and plot methods.
...	optional arguments. For coalescent.plot optional arguments include: sleep, the time to pause between generations (set to 0.2s by default); lwd, the line width for parent-offspring lines in the coalescent geneology; and col.order, if colors=NULL, whether to use 'sequential' (col.order="sequential", the default) or 'alternating' (col.order="alternating") colors for adjacent alleles.

Value

Creates a plot or animation.

Invisibly returns an object of class "coalescent.plot" containing the alleles (coded numerically) and the parent-offspring relationships from the coalescent simulation. This object can be printed or re-plotted using print and plot methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

drift.selection, genetic.drift

Examples

coalescent.plot()
	## Not run: 
		coalescent.plot(n=20,ngen=30,col.order="alternating")
		object<-coalescent.plot()
		print(object)
		plot(object)
	
## End(Not run)

---

Simulation of genetic drift & natural selection at a biallelic locus

Description

Simulates drift and natural selection at a single biallelic locus within one or various populations.

Usage

drift.selection(p0=0.5, Ne=100, w=c(1,1,1), ngen=400, nrep=10, colors=NULL, ...)

Arguments

p0	starting frequency for the A allele.
Ne	effective population size.
w	fitnesses of the three genotypes: AA, Aa, and aa.
nrep	number of replicate simulations.
ngen	total time, in number of generations, for the simulation.
colors	colors to use for plotting.
...	optional arguments. Presently the only arguments are type (e.g., "l", "s") and lwd in the plot method.

Value

The function creates a plot and returns an object of class "drift.selelction" consisting of list containing the allele frequency through time for each simulation. This object can be printed or plotted using corresponding methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

genetic.drift, selection

Examples

drift.selection()
	p<-drift.selection(p0=0.01,Ne=100,w=c(1,0.9,0.8),ngen=200,nrep=5)
	print(p)
	plot(p)

---

Simulation of a founder event or population bottleneck

Description

This function simulates genetic drift with a founding event / population bottleneck at time etime.

Usage

founder.event(p0=0.5, Ne=1000, Nf=10, ttime=100, etime=50, show="p", ...)

Arguments

p0	Starting frequency for the A allele.
Ne	Effective population size at the start of the simulation and after the founding event.
Nf	Size of the founding population.
ttime	Total time of the simulation.
etime	Time for the founding event. Can either be a single generation, or a sequence of generations (e.g., etime=40:50) for a prolonged founder event. Note that to simulate a prolonged bottleneck the user must supply the sequence of generations during which the population is to be bottlenecked and not merely the start and end times of the bottleneck.
show	Two different options for plotting. "p" shows the frequency of the A allele through time; "var" shows the genetic variation in the population, calculated as p*(1-p). The default is show="p".
...	optional arguments. Presently, the only optional argument in founder.event is ltype which specifies the line type and defaults to "s".

Value

The function creates one of two different plots, depending on the value of show.

The function also invisibly returns an object of class "founder.event" which can be printed or plotted using corresponding print and plot methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

drift.selection, genetic.drift

Examples

founder.event()
	p<-founder.event(show="variation")
	print(p)
	plot(p,show="p",ltype="l")

---

Numerical analysis of a frequency dependent seletion model

Description

This function performs numerical analysis of a frequency dependent selection model based on Rice (2004; Evolutionary Theory: Mathematical & Conceptual Foundations). The fitnesses of the three genotypes in the model are as follows, where f(XX) denotes the frequency of the XX genotype: w(AA)=1-3*f(Aa)+3*f(aa); w(Aa)=1-s*f(Aa); and w(aa)=1-3*f(Aa)+3*f(AA). As shown in Rice (2004), though entirely deterministic, the model can exhibit chaotic behavior under some values for s.

Usage

freqdep(p0=0.01, s=0, time=100, show="p", pause=0, ...)

Arguments

p0	Starting frequency for the A allele.
s	Parameter that determines the strength of selection against heterozygotes when they are common.
time	Number of generations to run the analysis.
show	Various options for plotting. "p" shows the frequency of A through time; "q" gives the frequency of the a allele; "fitness" gives the mean population fitness through time; "surface" plots the mean fitness as a function of p; "deltap" shows the change in p as a function of p; "cobweb" creates a cobweb plot showing p(t) by p(t+1). The default is show="p".
pause	Pause between generations. pause=0.01 (for instance) might smooth animation.
...	optional arguments. Presently, the only optional argument in freqdep is color, which can be used to change the color of the lines of the plot. The plot method can also accept the optional arguments type (e.g., "l" or "s") and lwd.

Value

The function creates one of several possible plots, depending on the value of show.

The use of cobweb plots follows selection.

The function also invisibly returns an object of class "freqdep" containing the frequency of the allele A through time, if this was calculated by the selected method. This can be printed or plotted using the corresponding methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

sexratio

Examples

freqdep(time=100)
	freqdep(s=1.5,time=100)
	p<-freqdep(s=2,show="cobweb",time=100)
	plot(p)

---

Genetic drift simulation

Description

This function simulates genetic drift at a biallelic genetic locus with no selection and no mutation in a sexually reproducing diploid population or set of populations. It is essentially redundant with drift.selection, but in which there is no difference in relative fitness among genotypes; however, it also allows the user to visualize heterozygosity or genetic variation through time - options that are not yet implemented in drift.selection.

Usage

genetic.drift(p0=0.5, Ne=20, nrep=10, time=100, show="p", pause=0.1, ...)

Arguments

p0	Starting frequency for the A allele.
Ne	Effective population size.
nrep	Number of replicate simulations.
time	Total time, in number of generations, for the simulation.
show	Various options for plotting. "p" shows the frequency of the A allele through time; "genotypes" creates an animated histogram with the frequencies of each of the three genotypes through time; "fixed" shows the fraction of populations that have fixed for each allele, a or A; "heterozygosity" plots the mean heterozygosity and the expected heterozygosity through time. The default is show="p".
pause	Pause between generations. pause=0.01 (for instance) might smooth animation.
...	optional arguments. In genetic.drift the optional arguments are presently: colors (a vector giving the colors to be used to graph the various simulations); and lwd. The plot method of the object class adds the optional argument type (e.g., "l" or "s".)

Value

The function creates one of several possible plots, depending on the value of show.

The function also invisibly returns an object of class "genetic.drift" that can be printed or re-plotted by the user using corresponding print and plot methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

drift.selection, founder.event, selection

Examples

## Not run: 
		genetic.drift()
		object<-genetic.drift(p0=0.7,show="heterozygosity")
		plot(object,show="genotypes")
	
## End(Not run)

---

Computes Hardy-Weinberg frequencies for a multiallelic locus or across multiple loci

Description

hardy.weinberg computes Hardy-Weinberg frequencies for a multiallelic locus using arbitrary allele frequencies.

multilocus.hw computes multilocus Hardy-Weinberg frequencies for a set of biallelic loci.

Usage

hardy.weinberg(p=c(0.5,0.5), alleles=c("A","a"), as.matrix=FALSE)
	multilocus.hw(nloci=2, p=NULL)

Arguments

p	allele frequencies. In the case of multilocus.hw the frequencies of the dominant (in this case, merely uppercase) allele at each locus.
alleles	names of the alleles.
as.matrix	logical argument indicating whether to return the result in the form of a matrix (if TRUE) or a vector.
nloci	for multilocus.hw the number of loci.

Value

Returns a matrix or vector.

Author(s)

Liam Revell [email protected]

See Also

phenotype.freq

Examples

hardy.weinberg()
	hardy.weinberg(p=c(0.4,0.3,0.2,0.1),alleles=letters[1:4])

---

Analysis of hawk-dove game theoretic model

Description

This function performs numerical analysis of a discrete-time hawk-dove model in which "payoff" determines relative fitness in the population.

Usage

hawk.dove(p=c(0.01,0.99), M=NULL, time=100)

Arguments

p	Starting frequency of hawk & dove phenotypes, respectively. Should correspond with the rows of M. If a single value is given then p will automatically be set to p=c(p,1-p).
M	Payoff matrix. M[i,j] should contain the fitness of i when interacting with j.
time	Number of generations.

Value

The function creates a two panel plot. The upper panel shows the relative frequencies of each of the two interacting phenotypes. The lower panel shows mean fitness of the population and of each morph through time.

The function also invisibly returns an object of class "hawk.dove" containing the frequencies of each strategy through time and their fitnesses. This object can be printed or re-plotted using corresponding print and plot methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

freqdep

Examples

hawk.dove(time=60)
	Payoff<-matrix(c(0.5,0.6,1.5,1.0),2,2)
	object<-hawk.dove(M=Payoff,time=60)
	print(object)
	plot(object)

---

Migration, drift, and selection

Description

Simulates migration, natural selection, and genetic drift. Selection can be in opposite directions in the two populations experiencing gene flow.

Usage

msd(p0=c(0.5,0.5), Ne=c(100,100), w=list(c(1,1,1),c(1,1,1)), 
		m=c(0.01,0.01), ngen=400, colors=c("red","blue"), ...)

Arguments

p0	starting frequency for the A allele in each of two populations.
Ne	effective population size for each of two populations.
w	fitnesses of the three genotypes (AA, Aa, and aa, in that order) in each of the two populations. w should take the form of a list of two vectors.
m	rates of migration from the first population to the second, and from the second population to the first, in that order. This value is best interpreted as the probability that an individual born in population 1 will migrate to population 2 before reproduction, and vice versa.
ngen	total time, in number of generations, for the simulation.
colors	colors to use for plotting.
...	optional arguments. Presently, the only optional argument for msd is show.legend (which defaults to TRUE). The plot method adds the additional optional arguments of colors (a vector of colors for the two simulated populations), lwd, and type (e.g., "l" or "s").

Value

The function creates a plot and invisibly returns a list containing the allele frequency through time for each of the two simulated populations.

The returned object is of class "msd" and can be printed or re-plotted using corresponding print or plot methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

drift.selection

Examples

msd()
	msd(p0=c(0.25,0.75),w=list(c(1,0.9,0.8),c(0.8,0.9,1)))
	object<-msd(p0=c(0.75,0.25),w=list(c(1,0.9,0.8),
		c(0.8,0.9,1)),m=c(0.1,0.1),ngen=100)
	print(object)
	plot(object,colors=c("black","grey"),lwd=4,type="s")

---

Gene frequencies over time under mutation-selection balance

Description

This function performs numerical analysis of mutation-selection balance with mutation from A to a and selection against (either or both of) Aa and aa.

Usage

mutation.selection(p0=1.0, w=c(1,0), u=0.001, time=100, show="q", pause=0, 
		ylim=c(0,1))

Arguments

p0	Starting frequency for the A allele.
w	Fitnesses of the heterozygote (Aa) and homozygote deleterious (aa) genotypes. The fitness of genotype AA is assumed to be 1.0.
u	Rate at which A alleles are converted to a alleles by mutation.
time	Number of generations to run the analysis.
show	Two options for plotting. "q" shows the frequency of a through time; "fitness" plots the mean fitness over time. The default is show="q".
pause	Pause between generations. pause=0.01 (for instance) might smooth animation.
ylim	Limits on the y-axis for plotting.

Value

The function creates one of three possible plots, depending on the value of show.

The function also invisibly returns the frequency of the A allele through time and the mean population fitness as an object of class "mutation.selection" that can be printed or re-plotted with associated print and plot methods, respectively. The plot method also permits user control over various attributes of the appearance of the plot, such as the color of the plotted lines (color), the line widths (lwd), the limits of the y-axis (ylim), and the type of line (e.g., "l" vs. "s", via the argument type).

Author(s)

Liam Revell [email protected]

See Also

genetic.drift, selection

Examples

mutation.selection(w=c(1,0),time=100,ylim=c(0,0.1))

---

Computes phenotypic distribution and its change through time due to natural selection on a polygenic trait

Description

phenotype.freq computes the phenotypic trait distribution for a polygenic trait. Can be used to demonstrate that the phenotypic distribution of a polygenic trait will tend to normality as the number of loci is increased, regardless of the allele frequencies at each locus.

phenotype.selection computes the change in the phenotypic trait distribution through time under natural selection. Can be used to show that natural selection on a polygenic trait can move the value of the trait well beyond its original distribution in the population.

Usage

phenotype.freq(nloci=6, p=NULL, effect=1/nloci)
	phenotype.selection(nloci=6, p=NULL, effect=1/nloci, beta=0.1, ngen=20, ...)

Arguments

nloci	number of loci. For simplicity all loci are assumed to be biallelic.
p	allele frequency, p, for each locus, in a vector. If not supplied, initially frequencies will be assumed to be 0.5 at all loci.
effect	additive effect of an allele substitution. For simplicity, this is assumed to be the same at all loci.
beta	selection gradient.
ngen	number of generations to analyze.
...	optional arguments. Presently the only optional argument in the function phenotype.selection is sleep, which can be used to specify the time delay in seconds between generations.

Value

Creates a plot or animation.

phenotype.freq also invisibly returns an object of class "phenotype.freq" that can be printed or re-plotted using print and plot methods corresponding to the object type. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

clt, selection

Examples

## Not run: 
		phenotype.freq(n=4)
		object<-phenotype.freq(nloci=6,p=runif(n=6),effect=1/6)
		print(object)
		plot(object)
		object<-phenotype.freq(nloci=10,p=runif(n=10),effect=rexp(n=10))
		print(object)
		plot(object)
		phenotype.selection(ngen=100)
	
## End(Not run)

---

Simulation of reproductive character displacement in an ecological community

Description

This function conducts individual-based, genetically explicit numerical simulation of reproductive character displacement in an ecological community. The model is one of multiple species (with fixed relative abundance) competing to utilize the same signal space. There is both stabilizing selection on the signal trait for detectability, as well as (in multi-species simulations) countervailing selection for divergence due to the costs of erroneous mismating attempts.

Usage

rcd(nsp=3, nindivs=c(700,400,100), w_t=10, gen=c(500,500), figs="on", pf=100, ...)

Arguments

nsp	Number of species in the simulation. If figs="on", nsp must be 1, 2, or 3.
nindivs	A vector of length nsp containing the integer number of individuals in each species of the simulation.
w_t	Shape parameter of the Gaussian selection surface for the male signalling trait.
gen	Vector containing the number of allopatric generations followed by the number of sympatric generations for simulation.
figs	Either "on" if plotting is turned on, or "off" to suppress plotting.
pf	Print frequency for the simulation status to screen.
...	Optional arguments.

Value

The function returns a list containing the mean male signal trait and the mean female preference over time. It also (optionally) plots these.

Author(s)

Liam Revell [email protected]

See Also

drift.selection, genetic.drift, freqdep, selection

Examples

## Not run: 
		obj<-rcd(nsp=2,nindivs=c(500,500))
	
## End(Not run)

---

Numerical analysis of biallelic locus frequency independent selection

Description

This function performs numerical analysis of a simple biallelic selection model.

Usage

selection(p0=0.01, w=c(1.0,0.9,0.8), time=100, show="p", pause=0, ...)

Arguments

p0	Starting frequency for the A allele.
w	Fitnesses for the three genotypes in the following order: AA, Aa, aa.
time	Number of generations to run the analysis.
show	Various options for plotting. "p" shows the frequency of A through time; "surface" plots the mean fitness as a function of p; "deltap" shows the change in p as a function of p; "cobweb" creates a cobweb plot showing p(t) by p(t+1). The default is show="p".
pause	Pause between generations. pause=0.01 (for instance) might smooth animation.
...	Optional arguments, including: add, a logical value indicating whether or not to add to the current plot (applies only to show="p"); color, change the color of the plotted line (works nicely with add, for obvious reasons); and equil, a logical value indicating whether or not to show the equilibrium value of p using vertical (or horizontal) lines on the graph (defaults to FALSE).

Value

The function creates one of several possible plots, depending on the value of show.

The cobweb plot shows p(t+1) as a function of p(t), with stairsteps giving the changes across generations given the initial value of p (p0) and total time (time) that are specified by the user.

The function invisibly returns an object of class "selection" which can be printed or re-plotted using associated print and plot methods. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

drift.selection, freqdep, msd, mutation.selection

Examples

selection(w=c(1.0,0.8,0.8),time=500)
	selection(w=c(1.0,1.0,0.0),show="surface")
	object<-selection(w=c(0.8,1.1,0.7))
	print(object)
	plot(object,show="cobweb")

---

Hypothetical analysis of frequency dependent selection on a sex determining genetic locus

Description

This function performs numerical analysis of a frequency dependent selection model of a hypothetical diploid sexually reproducing population in which sex is determined by the genotype at a biallelic genetic locus. Genotype AA are male, genotype aa are female, and genotype Aa might be male or female with probabilities that can be specified by the user. (Users may find, for instance, that setting sex.Aa=c(1,0) will result in evolution towards an XY sex determination system; whereas sex.Aa=c(0,1) will evolve towards a ZW system.)

Usage

sexratio(p0=0.01, time=40, show="p", pause=0, sex.Aa=c(0.5,0.5))

Arguments

p0	Starting frequency for the A allele. Individuals with AA genotypes are male, while individuals with Aa genotypes are male or female with probability given by sex.Aa.
time	Number of generations to run the analysis.
show	Two different options for plotting. "p" shows the frequency of A through time; "fitness" plots the mean fitness through time; "sex-ratio" plots the relative frequency of each sex; and "genotypes" plots the frequencies of the three genotypes in the population. The default is show="p".
pause	Pause between generations. pause=0.01 (for instance) might smooth animation.
sex.Aa	Probability that individuals with genotype Aa are male or female, respectively.

Value

The function creates one of four possible plots, depending on the value of show. Numerical analysis of this model shows how frequency dependent selection should favor alleles that tend to produce the rarer sex in the population.

The function invisibly returns an object of class "sexratio" that can be printed or re-plotted by the user. (See examples.)

Author(s)

Liam Revell [email protected]

See Also

freqdep

Examples

sexratio()
	sexratio(p0=0.001,show="sex-ratio")
	sexratio(p0=0.001,show="fitness")
	object<-sexratio(p0=0.001,sex.Aa=c(0.9,0.1),
		time=20)
	print(object)
	par(mfrow=c(2,1))
	plot(object,lwd=4,type="s",show="sex-ratio")
	plot(object,lwd=4,type="s",show="genotypes")
	par(mfrow=c(1,1))

---