zsm: Zero sum multinomial distribution as derived by McKane

Description Usage Arguments Value Note Author(s) References Examples

Description

The Zero sum multinomial distribution of species abundances as derived by McKane 2004.

Usage

1
zsm(J, P, m)

Arguments

J

Size of local community

P

Abundance in metacommunity

m

Probability of immigration

Value

Returns a vector of size J showing the probability of the stationary abundance being 0,1,2,...,J.

Note

The function uses lgamma() to avoid numerical overflow

Author(s)

Robin K. S. Hankin

References

A. J. McKane and others 2004. “Analytic solution of Hubbell's model of local community dynamics”. Theoretical Population Biology 65:67-73

Examples

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 sum(zsm(164,0.1,0.5))  # should be 1



# McKane et al 2004: figure 1.
layout(matrix(1:4,2,2))
par(mai=0.2+rep(0,4))
plot(1,type="n",log="y",ylim=c(1e-9,1),xlim=c(0,64),xlab="",ylab="Ps(N)",
     axes=FALSE,main=expression(J==64))
axis(1,pos=1e-9)
axis(2,pos=0,at=10^(-(0:9)))
segments(64,1e-9,64,1)
segments(60,1e-9,64,1e-9)
f <- function(P){points(0:64,zsm(64,P=P,m=0.05),type="l")}
for(i in 1:9){f(i/10)}
f(0.99)
f(0.999)
f(0.01)
f(0.001)
text(07,3.2e-7,adj=0,expression(P==0.999))
text(49,3.2e-7,adj=0,expression(P==0.001))
text(45,0.1,expression(m==0.05))


plot(1,type="n",log="y",ylim=c(1e-5,1),xlim=c(0,64),xlab="",ylab="Ps(N)",
     axes=FALSE,main="")
axis(1,pos=1e-5)
axis(2,pos=0,at=10^-(0:5))
segments(60,1e-5,64,1e-5)
segments(64,1e-5,64,1)
par(xpd=FALSE)
g <- function(m){points(0:64,pmax(zsm(64,P=0.1,m=m),1e-5),type="l")}
g(0.0001)
g(0.0005)
g(0.002)
g(0.01)
g(0.02)
g(0.05)
g(0.5)
g(0.999)
text(50,0.4,expression(P==0.1))

plot(1,type="n",log="y",ylim=c(1e-9,1),xlim=c(0,1e5),xlab="",ylab="Ps(N)",
     axes=FALSE,main=expression(J==10000))
axis(1,pos=1e-9)
axis(2,pos=0)
segments(1e5,1e-9,1e5,0.1)


h <- function(P){points(0:1e5,pmax(zsm(1e5,P=P,m=0.05),1e-9),type="l")}
for(i in 1:9){h(i/10)}
h(0.01)
h(0.99)
text(75000,0.1,expression(m==0.5))

plot(1,type="n",log="y",ylim=c(1e-40,1),xlim=c(0,1e5),xlab="",ylab="Ps(N)",
     axes=FALSE,main="")
axis(1,pos=1e-40)
axis(2,pos=0,at=1/10^c(40,32,24,16,8,0))
segments(1e5,1e-40,1e5,1)

i <- function(m){points(0:1e5,pmax(zsm(1e5,P=0.1,m=m),1e-40),type="l")}
i(0.0001)
i(0.0002)
i(0.0005)
i(0.001)
i(0.002)
i(0.005)
i(0.01)
i(0.02)
i(0.5)
text(60000,1e-4,expression(P==0.1))

untb documentation built on March 19, 2018, 9:03 a.m.