k_exponential_power {quaxnat}

Dispersal kernels from exponential power family

Description

k_exponential_power computes the value, multiplied by N, of a dispersal kernel from the exponential power family that includes, as special cases, Gaussian kernels and kernels that follow an exponential function of the distance.

Usage

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

which corresponds to a probability density of the distance given by

p(r)={b \over a^{d}\Gamma (d/b)}r^{d-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 Bateman (1947), Clark et al. (1998), Austerlitz et al. (2004), Nathan et al. (2012) for the planar case. This means the bth power of the distance has a gamma distribution with shape parameter d/b and scale parameter a^{b}.

The kernel has its maximum at zero and represents a rather flexible family that includes, for b=2 the classical Gaussian kernels and for b=1, kernels decreasing exponentially with the distance. For b<1 the distance distribution is fat-tailed in the sense of Kot et al. (1996). Such kernels have consequently been applied in a number of theoretical studies that address dispersal (Ribbens et al. 1994, Bullock et al. 2017).

Value

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

References

Bateman, A. (1947). Contamination in seed crops: III. relation with isolation distance. Heredity 1, 303–336. doi:10.1038/hdy.1947.20

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

Ribbens, E., Silander Jr, J.A., Pacala, S.W. (1994). Seedling recruitment in forests: calibrating models to predict patterns of tree seedling dispersion. Ecology 75, 1794–1806. doi:10.2307/1939638

Clark, J.S., Macklin, E., Wood, L. (1998). Stages and spatial scales of recruitment limitation in southern Appalachian forests. Ecological Monographs 68(2), 213–235. doi:10.2307/2657201

Clark, J.S. (1998). Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord. The American Naturalist 152(2), 204–224. doi:10.1086/286162

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

Bullock, J. M., Mallada González, L., Tamme, R., Götzenberger, L., White, S.M., Pärtel, M., Hooftman, D.A. (2017). A synthesis of empirical plant dispersal kernels. Journal of Ecology 105, 6–19. doi:10.1111/1365-2745.12666

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