This function allows to simulate genetic data from a non-equilibrium continent-island model, where each island can have a different size and a different inbreeding coefficient.
This function simulates genetic data under the continent-islands model (IIM=TRUE) or the finite island model (IIM=FALSE). In the IIM, a continent of infinite size sends migrants to islands of finite sizes \(N_i\) at a rate \(m\). Alleles can also mutate to a new state at a rate \(\mu\). Under this model, the expected \(F_{STi}, \theta_i\), can be calculated and compared to empirical estimates.
sim.genot.t(size=50,nbal=4,nbloc=5,nbpop=3,N=1000,
mig=0.001,mut=0.0001,f=0,t=100,IIM=TRUE)the number of sampled individuals per island
the number of alleles per locus (maximum of 99)
the number of loci to simulate
the number of islands to simulate
the effective population sizes of each island. If only one number, all islands are assumed to be of the same size
the migration rate from the continent to the islands
the mutation rate of the loci
the inbreeding coefficient for each island
the number of generation since the islands were created
whether to simulate a continent island Model (default) or a migrant pool island Model
A data frame with size*nbpop rows and nbloc+1 columns. Each row is an individual, the first column contains the island to which the individual belongs, the following nbloc columns contain the genotype for each locus.
In this model, \(\theta_t\) can be written as a function of population size \(N_i\), migration rate \(m\), mutation rate \(\mu\) and \(\theta_{(t-1)}\).
The rational is as follows:
With probability \(\frac{1}{N}\), 2 alleles from 2 different individuals in the current generation are sampled from the same individual of the previous generation:
-Half the time, the same allele is drawn from the parent;
-The other half, two different alleles are drawn, but they are identical in proportion \(\theta_{(t-1)}\).
-With probability \(1-\frac{1}{N}\), the 2 alleles are drawn from different individuals in the previous generation, in which case they are identical in proportion \(\theta_{(t-1)}\).
This holds providing that neither alleles have mutated or migrated. This is the case with probability \((1-m)^2 \times (1-\mu)^2\). If an allele is a mutant or a migrant, then its coancestry with another allele is 0 in the infinite continent-islands model (it is not the case in the finite island model).
Note also that the mutation scheme assumed is the infinite allele (or site) model. If the number of alleles is finite (as will be the case in what follows), the corresponding mutation model is the K-allele model and the mutation rate has to be adjusted to \(\mu'=\frac{K-1}{K}\mu\).
Lets substitute \(\alpha\) for \((1-m)^2 (1-\mu)^2\) and \(x\) for \(\frac{1}{2N}\).
The expectation of \(F_{ST}\), \(\theta\) can be written as:
$$\theta_t=(\alpha (1-x))^t \theta_0 + \frac{x}{1-x}\sum_{i=1}^t (\alpha (1-x))^i$$
which reduces to \(\theta_t=\frac{x}{1-x}\sum_{i=1}^t (\alpha (1-x))^i\) if \(\theta_0=0\).
Transition equations for \(theta\) in the migrant-pool island model (IIM=FALSE) are given in Rouseet (1996). Currently, the migrant pool is made of equal contribution from each island, irrespective of their size.
Rousset, F. (1996) Equilibrium values of measures of population subdivision for stepwise mutation processes. Genetics 142:1357
# NOT RUN {
psize<-c(100,1000,10000,100000,1000000)
dat<-sim.genot.t(nbal=4,nbloc=20,nbpop=5,N=psize,mig=0.001,mut=0.0001,t=100)
summary(wc(dat)) #Weir and cockerham overall estimators of FST & FIS
betas(dat) # Population specific estimator of FST
# }
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