0th

Percentile

Various functions from Vallade and Houchmandzadeh (2003), dealing with analytical solutions of a neutral model of biodiversity

Keywords
math
Usage
vallade.eqn5(JM, theta, k)
vallade.eqn12(J, omega, m, n)
vallade.eqn14(J, theta, m, n)
vallade.eqn17(mu, theta, omega, give=FALSE)
Arguments
J,JM
Size of the community and metacommunity respectively
theta
Biodiversity number $theta=(JM-1)nu/(1-nu)$ as discussed in equation 6
k,n
Abundance
omega
Relative abundance $k/JM$
m
Immigration probability
mu
Scaled immigration probability $mu=(J-1)m/(1-m)$
give
In function vallade.eqn17(), Boolean with default FALSE meaning to return the numerical value of the integral and TRUE meaning to return the entire output of integrate() including the error estimates
Details

Notation follows Vallade and Houchmandzadeh (2003) exactly.

Note

Function vallade.eqn16() requires the polynom library, which is not loaded by default. It will not run for $J>50$ due to some stack overflow error.

Function vallade.eqn5() is identical to function alonso.eqn6()

References

M. Vallade and B. Houchmandzadeh 2003. “Analytical Solution of a Neutral Model of Biodiversity”, Physical Review E, volume 68. doi: 10.1103/PhysRevE.68.061902

Examples
# A nice check:
JM <- 100
k <- 1:JM
sum(k*vallade.eqn5(JM,theta=5,k))  # should be JM=100 exactly.

# Now, a replication of Figure 3:
omega <- seq(from=0.01, to=0.99,len=100)
f <- function(omega,mu){
}
plot(omega,
omega*5,type="n",xlim=c(0,1),ylim=c(0,5),
xlab=expression(omega),
ylab=expression(omega*g[C](omega)),
points(omega,omega*sapply(omega,f,mu=0.5),type="l")
points(omega,omega*sapply(omega,f,mu=1),type="l")
points(omega,omega*sapply(omega,f,mu=2),type="l")
points(omega,omega*sapply(omega,f,mu=4),type="l")
points(omega,omega*sapply(omega,f,mu=8),type="l")
points(omega,omega*sapply(omega,f,mu=16),type="l")
points(omega,omega*sapply(omega,f,mu=Inf),type="l")

# Now a discrete version of Figure 3 using equation 14:
J <- 100
omega <- (1:J)/J

f <- function(n,mu){
m <- mu/(J-1+mu)
vallade.eqn14(J=J, theta=5, m=m, n=n)
}
plot(omega,omega*0.03,type="n",main="Discrete version of Figure 3 using
eqn 14")
points(omega,omega*sapply(1:J,f,mu=16))
points(omega,omega*sapply(1:J,f,mu=8))
points(omega,omega*sapply(1:J,f,mu=4))
points(omega,omega*sapply(1:J,f,mu=2))
points(omega,omega*sapply(1:J,f,mu=1))
points(omega,omega*sapply(1:J,f,mu=0.5))


Documentation reproduced from package untb, version 1.7-2, License: GPL

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