k_weibull {quaxnat}

Dispersal kernels for Weibull distance distributions

Description

k_weibull computes the value, multiplied by N, of the dispersal kernel from Tufto et al. (1997) based on seeds having a distance with a Weibull distribution from their source.

Usage

k_weibull(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 scale and shape parameters a and b of the distance distribution.

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

which corresponds to a probability density of the distance given by

p(r)={b \over a^{b}}r^{b-1}e^{-(r/a)^{b}},

where d is the spatial dimension, \left\|{\,}\right\| denotes the Euclidean norm and the normalizing constants involve the gamma function; see Tufto et al. (1997) for the planar case. Thus, the distance is assumed to have the Weibull distribution with scale parameter a and shape parameter b. Equivalently, the bth power of the distance has an exponential distribution with scale parameter a^{b}.

Consequently, if and only if b<1, the distance distribution has a heavier tail than an exponential distribution, although with tail probabilities still decreasing faster than any power law; it is a fat-tailed distribution in the sense of Kot et al. (1996). The kernel coincides with a Gaussian kernel in the special case b=d=2.

Value

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

References

Tufto, J., Engen, S., Hindar, K. (1997). Stochastic dispersal processes in plant populations, Theoretical Population Biology 52(1), 16–26. doi:10.1006/tpbi.1997.1306

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

Kot, M., Lewis, M.A., van den Driessche, P. (1996). Dispersal Data and the Spread of Invading Organisms. Ecology 77(7), 2027–2042. doi:10.2307/2265698

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

Examples

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