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Various functions from Vallade and Houchmandzadeh

vallade.Rd

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

Usage

vallade.eqn5(JM, theta, k)
vallade.eqn7(JM, theta)
vallade.eqn12(J, omega, m, n)
vallade.eqn14(J, theta, m, n)
vallade.eqn16(J, theta, mu)
vallade.eqn17(mu, theta, omega, give=FALSE)

Arguments

J,JM

Size of the community and metacommunity respectively

theta

Biodiversity number \(\theta=(J_M-1)\nu/(1-\nu)\) as discussed in equation 6

k,n

Abundance

omega

Relative abundance \(\omega=k/J_M\)

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.

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

Author

Robin K. S. Hankin

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()

Examples

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



# Now, a replication of Figure 3:
  omega <- seq(from=0.01, to=0.99,len=100)
  f <- function(omega,mu){
    vallade.eqn17(mu,theta=5, omega=omega)
  }
  plot(omega,
  omega*5,type="n",xlim=c(0,1),ylim=c(0,5),
        xlab=expression(omega),
        ylab=expression(omega*g[C](omega)),
        main="Figure 3 of Vallade and Houchmandzadeh")
  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))