k_lognormal {quaxnat}

Dispersal kernels for log-normal distance distributions

Description

k_lognormal computes the value, multiplied by N, of a dispersal kernel based on seeds having a distance with a log-normal distribution from the their source.

Usage

k_lognormal(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 log-transformed scale and shape parameters, given by the median distance a and by the variance b of the underlying normal 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)={\Gamma (d/2) \over 2\pi ^{d/2}\left\|{x}\right\|^{d}\sqrt{2\pi b}} e^{-{1 \over 2b}(\log (\left\|{x}\right\|/a))^{2}} ={\Gamma (d/2)e^{d^{2}b/2} \over 2\pi ^{d/2}a^{d}\sqrt{2\pi b}} e^{-{1 \over 2b}(\log {\left\|{x}\right\| \over a}+db)^{2}},

which corresponds to a probability density of the distance given by

p(r)={1 \over r\sqrt{2\pi b}}e^{-{1 \over 2b}(\log (r/a))^{2}} ={e^{b/2} \over a\sqrt{2\pi b}} e^{-{1 \over 2b}(\log {r \over a}+b)^{2}},

where d is the spatial dimension, \left\|{\,}\right\| denotes the Euclidean norm and the normalizing constant of the kernel involves the gamma function; see Greene and Johnson (1989), Stoyan and Wagner (2001) for the planar case. Thus, the distance is assumed to have the log-normal distribution such that the log-distance has a normal distribution with mean \log a and variance b. Here \log k(x) is a quadratic function of \log \left\|{x}\right\| with a maximum at \log a-db, while \log p(r) is a quadratic function of \log r with a maximum at \log a-b.

This kernel is particularly suitable if the maximum regeneration density is not directly at the seed source (e.g. Janzen–Connell effect), cf. Nathan et al. (2012).

Value

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

References

Greene, D.F., Johnson, E.A. (1989). A model of wind dispersal of winged or plumed seeds. Ecology 70(2), 339–347. doi:10.2307/1937538

Stoyan, D., Wagner, S. (2001). Estimating the fruit dispersion of anemochorous forest trees. Ecol. Modell. 145, 35–47. doi:10.1016/S0304-3800(01)00385-4

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