k_power {quaxnat}

Power-law dispersal kernels

Description

k_power computes the value, multiplied by N, of a dispersal kernel that follows a power law of a constant a plus the distance.

Usage

k_power(x, par, N = 1, d = NCOL(x))

Arguments

x

Numeric matrix of positions x relative to the seed source, or vector of distances \left\|{x}\right\| to the seed source.

par

Numeric vector with two elements representing the log-transformed parameters a and b.

N

The multiplier N.

d

The spatial dimension.

Details

The dispersal kernel, i.e. spatial probability density of the location of a seed relative to its source, is here given by

k(x)={\Gamma (d/2) \over 2\pi ^{d/2}a^{d}B(d,b)} (1+{\left\|{x}\right\| \over a})^{-(b+d)},

which corresponds to a probability density of the distance given by

p(r)={1 \over a^{d}B(d,b)}r^{d-1}(1+{r \over a})^{-(b+d)},

where d is the spatial dimension, \left\|{\,}\right\| denotes the Euclidean norm and the normalizing constants involve the beta and gamma functions; see Nathan et al. (2012) for the planar case (with b replaced by b-d). This means the distance is da \over b times a random variable having an F distribution with 2d and 2b degrees of freedom. This is a fat-tailed distribution for all choices of the parameter b.

Value

Numeric vector of function values k(x) multiplied by N.

References

Nathan, R., Klein, E., Robledo‐Arnuncio, J.J., Revilla, E. (2012). Dispersal kernels: review, in Clobert, J., Baguette, M., Benton, T.G., Bullock, J.M. (eds.), Dispersal ecology and evolution, 186–210. doi:10.1093/acprof:oso/9780199608898.003.0015

Austerlitz, F., Dick, C.W., Dutech, C., Klein, E.K., Oddou-Muratorio, S., Smouse, P.E., Sork, V.L. (2004). Using genetic markers to estimate the pollen dispersal curve. Molecular Ecology 13, 937–954. doi:10.1111/j.1365-294X.2004.02100.x

Examples

k_power(2:5, par=c(0,0), d=2)