spatialwarnings (version 3.1.0)

dLSW: The Lifshitz-Slyozov-Wagner distribution

Description

Density and distribution function for the Lifshitz-Slyozov-Wagner (LSW) distribution with mean mu.

Usage

dLSW(x, mu, log = FALSE)
pLSW(x, mu, lower.tail = TRUE, log.p = FALSE)
LSW_fit(x)

Value

dLSW gives the density, pLSW gives the distribution function, both as numerical vectors determined by the length of x.

Arguments

x

vector of quantiles

mu

the mean of the distribution

log, log.p

logical; if TRUE, probabilities p are given as log(p)

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise, \(P[X > x]\)

Details

The LSW distribution is a continuous distribution with density

$$ f(x) = \frac{4x^2}{9\mu^3} ( \frac{3\mu}{3\mu + x} )^{7/3} ( \frac{3\mu}{3\mu - 2x} )^{11/3} e^{\frac{2x}{2x - 3\mu}} $$

where \(\mu\) is the mean of the distribution.

The functions dLSW gives the probability density, pLSW gives the distribution function. qLSW and rLSW are not implemented. You can use LSW_fit to fit an LSW distribution to a set of observations.

The length of the results is determined by the length of x, and mu can only be a single value.

Please note that this distribution has support only on the interval [0,mu*3/2). Probabilities outside this interval are returned as 0.

References

Siteur, Koen, Quan-Xing Liu, Vivi Rottschäfer, Tjisse van der Heide, Max Rietkerk, Arjen Doelman, Christoffer Boström, and Johan van de Koppel. 2023. "Phase-Separation Physics Underlies New Theory for the Resilience of Patchy Ecosystems." Proceedings of the National Academy of Sciences 120 (2): e2202683120. https://doi.org/10.1073/pnas.2202683120.

See Also

lsw_sews

Examples

# Plot the density 
x <- seq(0, 10, l = 128) 
plot(x, dLSW(x, mu = 3), type = "l", col = "black")
lines(x, dLSW(x, mu = 5), type = "l", col = "red")
lines(x, dLSW(x, mu = 7), type = "l", col = "blue")
legend(x = 0, y = max(dLSW(x, mu = 3)), lty = 1, col = c("black", "red", "blue"), 
       legend = paste("mu =", c(3, 5, 7)))