turnover: Specifying a Dynamic Population

Description

sim.popn can simulate a multi-session population with known between-session survival, recruitment and movement probabilities. The parameter settings to achieve this are passed to sim.popn in its `details' argument. Components of `details' that are relevant to turnover are described below; see sim.popn for others.

Multi-session populations are generated in sim.popn whenever its argument `nsessions' is greater than 1. If details$lambda remains NULL (the default) then the population for each successive session is generated de novo from the given density model (model2D, D etc.). If a value is specified for details$lambda then only the first population is generated de novo; remaining populations are generated iteratively with probabilistic mortality, recruitment and movement as described here.

Turnover components of <code>sim.popn</code> details argument

Component	Description	Default
`phi`	per capita survival rate $\phi$	0.7
`survmodel`	probability model for number of survivors	``binomial"
`lambda`	finite rate of increase $\lambda = N_{t+1} / N_t$	none
`recrmodel`	probability model for number of recruits	``poisson"
`superN`	optional superpopulation size for multinomial recruitment model	see below
`sigma.m`	spatial scale of movement kernel	0
`edgemethod`	treatment of animals that cross the boundary	``wrap"
`wrap`	deprecated in 3.1.6; use edgemethod	TRUE i.e. edgemethod = ``wrap"

Survival

Survival is usually thought of as a Bernoulli process (outcome 0 or 1 for each individual) so the number of survivors $S$ is a binomial variable (survmodel = "binomial"). Another approach is to fix the proportion surviving, but this can be done exactly only when $\phi N$ is an integer. A (slightly ad hoc) solution is to randomly choose between the two nearest integers with probability designed in the long term (over many sessions) to give the required $\phi$ (survmodel = "discrete").

Population growth and recruitment

Per capita recruitment ($f$) is the difference between lambda and phi ($f = \lambda - \phi$), which must be non-negative (phi > lambda causes an error). The number of recruits B is a random variable whose probability distribution is controlled by details$recrmodel:

Value	Probability model
"constantN"	Exact replacement of animals that die (B = $N_t - S$)
"binomial"	Binomial number of recruits (B ~ bin($N_t, f$)
"poisson"	Poisson number of recruits (B ~ pois($f N_t$))
"discrete"	Minimum-variance number of recruits (see Survival)
"multinomial"	The POPAN model, conditioning on superpopulation size (e.g., Schwarz and Arnason 1996))

In the case of binomial recruitment there is a maximum of one recruit per existing individual, so lambda <= (phi+1). Multinomial recruitment requires a value for the superpopulation size. This may be provided as the details component "superN". If not specified directly, a value is inferred by projecting a trial initial (simulated) population using the specified phi and lambda.

Movement

Individuals may shift their home range centre between sessions. Movement probability is governed by a circular bivariate normal (Gaussian) kernel with scale sigma.m.

sigma.m is also used to indicate two special cases:

sigma.m = 0 indicates no change in home range centres between sessions.

sigma.m < 0 causes a new home range centre for each animal to be drawn at random from within the buffered area.

If movement takes an animal across the boundary of the arena in sim.popn the component "edgemethod" comes into play. By default, locations are toroidally wrapped i.e. the animal re-joins the population on the opposing edge. Other options are ``clip'' (discard) and ``none" (allow centres outside the buffered area).

Examples

Run this code

# NOT RUN {
par (mfrow = c(2,3), mar = c(1,1,1,1))

## birth and death only
grid <- make.grid(nx = 7, ny = 4, detector = 'proximity', spacing = 10)
pop <- sim.popn (Nbuffer = 100, core = grid, nsessions = 6,    
    details = list(lambda = 0.8, phi = 0.6, sigma.m = 0))
sapply(pop, nrow)  ## how many individuals?
plot(pop)

## movement only
pop2 <- sim.popn (Nbuffer = 100, core = grid, nsessions = 6,    
    details = list(lambda = 1, phi = 1, sigma.m = 10, wrap = TRUE))
pop3 <- sim.popn (Nbuffer = 100, core = grid, nsessions = 6,    
    details = list(lambda = 1, phi = 1, sigma.m = 10, wrap = FALSE))
sapply(pop2, nrow)  ## how many individuals?
plot(pop2)

## show effect of toroidal wrapping --
tracks <- function (pop, ...) {
    rn <- unlist(lapply(pop, rownames))
    plot(pop[[1]], pch = 16)
    for (i in unique(rn))
       lines(t(sapply(pop, '[', i, 1:2)), ...)
}
par (mfrow=c(1,2))
tracks(pop2, type = 'o')
tracks(pop3, type = 'o')

# }