vallade

From untb v1.3-3 by Robin S. Hankin

Various functions from Vallade and Houchmandzadeh

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.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 e

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){
    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))

Documentation reproduced from package untb, version 1.3-3, License: GPL

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