Beverton-Holt settler-recruit relationship

Description

Calculates recruitment based on the settler-recruit relationship from Beverton & Holt (1957): slope * settlers / (1+slope*settlers/Rmax)

Usage

BevertonHolt(S, slope = 1/0.35, Rmax = 1)

Arguments

Details

slope and Rmax can both either be scalars or vectors of the same length as S.

Value

A vector of recruitment values.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Beverton RJH, Holt SJ (1957) On the dynamics of exploited fish populations. H.M.S.O., London. 533 pp.

Tools for working with connectivity matrices

Description

Tools for Working with Connectivity Data

Details

Collects several different methods for analyzing and working with connectivity data in R. Though primarily oriented towards marine larval dispersal, many of the methods are general and useful for terrestrial systems as well.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

Marco Andrello marco.andrello@gmail.com

References

Jacobi, M. N., and Jonsson, P. R. 2011. Optimal networks of nature reserves can be found through eigenvalue perturbation theory of the connectivity matrix. Ecological Applications, 21: 1861-1870.

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

Gruss, A., Kaplan, D. M., and Lett, C. 2012. Estimating local settler-recruit relationship parameters for complex spatially explicit models. Fisheries Research, 127-128: 34-39.

Kaplan, D. M., Botsford, L. W., and Jorgensen, S. 2006. Dispersal per recruit: An efficient method for assessing sustainability in marine reserve networks. Ecological Applications, 16: 2248-2263.

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

White, J. W. 2010. Adapting the steepness parameter from stock-recruit curves for use in spatially explicit models. Fisheries Research, 102: 330-334.

Gruss A, Kaplan DM, Hart DR (2011) Relative Impacts of Adult Movement, Larval Dispersal and Harvester Movement on the Effectiveness of Reserve Networks. PLoS ONE 6:e19960

Beverton RJH, Holt SJ (1957) On the dynamics of exploited fish populations. H.M.S.O., London. 533 pp.

See Also

See optimalSplitConnMat, d.rel.conn.beta.prior

Examples

## Not run: optimalSplitConnMat(CM)

Extended DPR population dynamics model to include homerange movement

Description

This function implements the marine population dynamics model described in Gruss et al. (2011). The model is an extension of the dispersal-per-recruit model in Kaplan et al. (2006) to include movement in a homerange and a gravity model for fishing effort redistribution.

Usage

DPRHomerangeGravity(
  larval.mat,
  adult.mat,
  recruits0,
  f0,
  timesteps = 10,
  settler.recruit.func = hockeyStick,
  LEP.of.f = function(f) 1 - f,
  YPR.of.f = function(f) f,
  gamma = 0,
  gravity.ts.interval = 1,
  ...
)

Arguments

Value

A list with the following elements:

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Gruss A, Kaplan DM, Hart DR (2011) Relative Impacts of Adult Movement, Larval Dispersal and Harvester Movement on the Effectiveness of Reserve Networks. PLoS ONE 6:e19960

Kaplan, D. M., Botsford, L. W., and Jorgensen, S. 2006. Dispersal per recruit: An efficient method for assessing sustainability in marine reserve networks. Ecological Applications, 16: 2248-2263.

See Also

See also BevertonHolt, hockeyStick, DispersalPerRecruitModel

Population dynamics model based on lifetime-egg-production

Description

This function implements the marine population dynamics model described in Kaplan et al. (2006). This model is most appropriate for examining equilibrium dynamics of age-structured populations or temporal dynamics of semelparous populations.

Usage

DispersalPerRecruitModel(
  LEP,
  conn.mat,
  recruits0,
  timesteps = 10,
  settler.recruit.func = hockeyStick,
  ...
)

Arguments

Value

A list with the following elements:

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan, D. M., Botsford, L. W., and Jorgensen, S. 2006. Dispersal per recruit: An efficient method for assessing sustainability in marine reserve networks. Ecological Applications, 16: 2248-2263.

See Also

See also BevertonHolt, hockeyStick

Examples

library(ConnMatTools)
data(chile.loco)

# Get appropriate collapse slope
# critical.FLEP=0.2 is just an example
slope <- settlerRecruitSlopeCorrection(chile.loco,critical.FLEP=0.2)

# Make the middle 20 sites a reserve
# All other sites: scorched earth
n <- dim(chile.loco)[2]
LEP <- rep(0,n)
nn <- round(n/2)-9
LEP[nn:(nn+19)] <- 1

Rmax <- 1

recruits0 <- rep(Rmax,n)

# Use DPR model
ret <- DispersalPerRecruitModel(LEP,chile.loco,recruits0,1:20,slope=slope,Rmax=Rmax,
                                settler.recruit.func=BevertonHolt)
image(1:n,1:20,ret$settlers,xlab="sites",ylab="timesteps",
      main=c("Settlement","click to proceed"))
locator(1)
plot(ret$settlers[,20],xlab="sites",ylab="equilibrium settlement",
     main="click to proceed")
locator(1)

# Same, but with a uniform Laplacian dispersal matrix and hockeyStick
cm <- laplacianConnMat(n,10,0,"circular")
ret <- DispersalPerRecruitModel(LEP,cm,recruits0,1:20,slope=1/0.35,Rmax=Rmax)
image(1:n,1:20,ret$settlers,xlab="sites",ylab="timesteps",
      main=c("Settlement","click to proceed"))
locator(1)
plot(ret$settlers[,20],xlab="sites",ylab="equilibrium settlement")

Compute vector of beta values

Description

Helper function to compute a set of beta values using formula used in Jacobi et al. (2012).

Usage

betasVectorDefault(n, steps = 10, cycles = 3/4, coeff = 0.8, pwr = 3)

Arguments

Value

vector of beta values

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

See Also

See also optimalSplitConnMat

Connectivity matrix for loco (Concholepas concholepas) from Chile

Description

Sample connectivity matrix representing potential larval dispersal of loco (Concholepas concholepas) from Chile. The matrix is for 89 sites along the coast of Chile and is derived from a theoretical larval transport model.

Format

A square 89x89 matrix with real, positive elements.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Garavelli L, Kaplan DM, Colas F, Stotz W, Yannicelli B, Lett C (2014) Identifying appropriate spatial scales for marine conservation and management using a larval dispersal model: The case of Concholepas concholepas (loco) in Chile. Progress in Oceanography 124:42-53. doi:10.1016/j.pocean.2014.03.011

Returns probability density function (PDF) for a mix of marked and unmarked individuals

Description

This function returns a probability density function (PDF) for scores for a mix of marked and unmarked individuals with known fraction of marked individuals. The distributions for marked individuals and for unmarked individuals must be known.

Usage

d.mix.dists.func(d.unmarked, d.marked)

Arguments

Value

A function representing the PDF of observations drawn from the mixed distribution of marked and unmarked individuals. The function takes two arguments: p.marked, the fraction of marked individuals in the distribution; and obs, a vector of observed score values.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

See also d.rel.conn.dists.func, optim.rel.conn.dists.

Estimate the probability distribution of relative connectivity values assuming a Beta-distributed prior

Description

These functions calculate the probability density function (d.rel.conn.beta.prior), the probability distribution function (aka the cumulative distribution function; p.rel.conn.beta.prior) and the quantile function (q.rel.conn.beta.prior) for the relative (to all settlers at the destination site) connectivity value for larval transport between a source and destination site given a known fraction of marked individuals (i.e., eggs) in the source population. A non-uniform prior is used for the relative connectivity value.

Usage

d.rel.conn.beta.prior(
  phi,
  p,
  k,
  n,
  prior.shape1 = 0.5,
  prior.shape2 = prior.shape1,
  prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2),
  ...
)

p.rel.conn.beta.prior(
  phi,
  p,
  k,
  n,
  prior.shape1 = 0.5,
  prior.shape2 = prior.shape1,
  prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2),
  ...
)

q.rel.conn.beta.prior.func(
  p,
  k,
  n,
  prior.shape1 = 0.5,
  prior.shape2 = prior.shape1,
  prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2),
  N = 1000,
  ...
)

q.rel.conn.beta.prior(
  q,
  p,
  k,
  n,
  prior.shape1 = 0.5,
  prior.shape2 = prior.shape1,
  prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2),
  N = 1000,
  ...
)

Arguments

Details

The prior distribution for relative connectivity phi defaults to a Beta distribution with both shape parameters equal to 0.5. This is the Reference or Jeffreys prior for a binomial distribution parameter. Both shape parameters equal to 1 corresponds to a uniform prior.

Estimations of the probability distribution are based on numerical integration using the integrate function, and therefore are accurate to the level of that function. Some modification of the default arguments to that function may be necessary to acheive good results for certain parameter values.

Value

Vector of probabilities or quantiles, or a function in the case of q.rel.conn.beta.prior.func.

Functions

• d.rel.conn.beta.prior: Returns the probability density for relative connectivity between a pair of sites

• p.rel.conn.beta.prior: Returns the cumulative probability distribution for relative connectivity between a paire of sites

• q.rel.conn.beta.prior.func: Returns a function to estimate quantiles for the probability distribution function for relative connectivity between a pair of sites.

• q.rel.conn.beta.prior: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multinomial.unnorm(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), dual.mark.transmission(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)

k <- 10 # Number of marked settlers among sample
n.obs <- 87 # Number of settlers in sample

p <- 0.4 # Fraction of eggs that was marked
phi <- seq(0.001,1-0.001,length.out=101) # Values for relative connectivity

# Probability distribution assuming infinite settler pool and uniform prior
drc <- d.rel.conn.unif.prior(phi,p,k,n.obs)
qrc <- q.rel.conn.unif.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval

# Probability distribution assuming infinite settler pool and using reference/Jeffreys prior
drp <- d.rel.conn.beta.prior(phi,p,k,n.obs)
prp <- p.rel.conn.beta.prior(phi,p,k,n.obs)
qrp <- q.rel.conn.beta.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval

# Make a plot of different distributions
# black = Jeffreys prior; red = uniform prior
# Jeffreys prior draws distribution slightly towards zero
plot(phi,drp,type="l",main="Probability of relative connectivity values",
     xlab=expression(phi),ylab="Probability density")
lines(phi,drc,col="red")
abline(v=qrp,col="black",lty="dashed")
abline(v=qrc,col="red",lty="dashed")

Functions for estimating the probability distribution for relative connectivity given a pair of distributions for scores for marked and unmarked individuals

Description

These functions return functions that calculate the probability density function (d.rel.conn.dists.func), the probability distribution function (aka the cumulative distribution function; p.rel.conn.dists.func) and the quantile function (q.rel.conn.dists.func) for relative connectivity given a set of observed score values, distributions for unmarked and marked individuals, and an estimate of the fraction of all eggs marked at the source site, p.

Usage

d.rel.conn.dists.func(
  obs,
  d.unmarked,
  d.marked,
  p = 1,
  N = max(100, min(5000, 2 * length(obs))),
  prior.shape1 = 0.5,
  prior.shape2 = prior.shape1,
  prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2),
  ...
)

p.rel.conn.dists.func(
  obs,
  d.unmarked,
  d.marked,
  p = 1,
  N = max(100, min(5000, 2 * length(obs))),
  prior.shape1 = 0.5,
  prior.shape2 = prior.shape1,
  prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2),
  ...
)

q.rel.conn.dists.func(
  obs,
  d.unmarked,
  d.marked,
  p = 1,
  N = max(100, min(5000, 2 * length(obs))),
  prior.shape1 = 0.5,
  prior.shape2 = prior.shape1,
  prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2),
  ...
)

Arguments

Details

The normalization of the probability distribution is carried out using a simple, fixed-step trapezoidal integration scheme. By default, the number of steps between relative connectivity values of 0 and 1 defaults to 2*length(obs) so long as that number is comprised between 100 and 5000.

Value

A function that takes one argument (the relative connectivity for d.rel.conn.dists.func and p.rel.conn.dists.func; the quantile for q.rel.conn.dists.func) and returns the probability density, cumulative probability or score value, respectively. The returned function accepts both vector and scalar input values.

Functions

• d.rel.conn.dists.func: Returns a function that is PDF for relative connectivity

• p.rel.conn.dists.func: Returns a function that is cumulative probability distribution for relative connectivity

• q.rel.conn.dists.func: Returns a function that is quantile function for relative connectivity

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.finite.settlement(), d.rel.conn.multinomial.unnorm(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), dual.mark.transmission(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)
data(damselfish.lods)

# Histograms of simulated LODs
l <- seq(-1,30,0.5)
h.in <- hist(damselfish.lods$in.group,breaks=l)
h.out <- hist(damselfish.lods$out.group,breaks=l)

# PDFs for marked and unmarked individuals based on simulations
d.marked <- stepfun.hist(h.in)
d.unmarked <- stepfun.hist(h.out)

# Fraction of adults genotyped at source site
p.adults <- 0.25

# prior.shape1=1 # Uniform prior
prior.shape1=0.5 # Jeffreys prior

# Fraction of eggs from one or more genotyped parents
p <- dual.mark.transmission(p.adults)$p

# PDF for relative connectivity
D <- d.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Estimate most probable value for relative connectivity
phi.mx <- optim.rel.conn.dists(damselfish.lods$real.children,
                                    d.unmarked,d.marked,p)$phi

# Estimate 95% confidence interval for relative connectivity
Q <- q.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Plot it up
phi <- seq(0,1,0.001)
plot(phi,D(phi),type="l",
     xlim=c(0,0.1),
     main="PDF for relative connectivity",
     xlab=expression(phi),
     ylab="Probability density")

abline(v=phi.mx,col="green",lty="dashed")
abline(v=Q(c(0.025,0.975)),col="red",lty="dashed")

Estimate the probability distribution for the number of settlers originating at a site given a sample from a finite settler pool

Description

These functions calculate the probability mass function (d.rel.conn.finite.settlement), the cumulative distribution function (p.rel.conn.finite.settlement) and the quantile function (q.rel.conn.finite.settlement) for the true number of settlers at a site that originated in a particular site given a known fraction of marked eggs among the eggs originating at the source site, a sample of settlers at the destination site, a known fraction of which are marked, and a finite settler pool of known size.

Usage

d.rel.conn.finite.settlement(
  n.origin,
  p,
  k,
  n.obs,
  n.settlers,
  prior.n.origin = 1
)

p.rel.conn.finite.settlement(
  n.origin,
  p,
  k,
  n.obs,
  n.settlers,
  prior.n.origin = 1
)

q.rel.conn.finite.settlement(q, p, k, n.obs, n.settlers, prior.n.origin = 1)

Arguments

Details

The relative connectivity between the source and destination sites is calculated as n.origin/n.settlers.

Value

A vector of probabilities or quantiles.

Functions

• d.rel.conn.finite.settlement: Returns the probability mass function for the numbers of settlers in the cohort that originated at the source site (i.e., site of marking).

• p.rel.conn.finite.settlement: Returns the cumulative distribution function for the numbers of settlers in the cohort that originated at the source site (i.e., site of marking).

• q.rel.conn.finite.settlement: Returns quantiles of the cumulative distribution function for the numbers of settlers in the cohort that originated at the source site (i.e., site of marking).

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.multinomial.unnorm(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), dual.mark.transmission(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)

k <- 10 # Number of marked settlers among sample
n.obs <- 87 # Number of settlers in sample
n.settlers <- 100 # Total size of settler pool

p <- 0.4 # Fraction of eggs that was marked
phi <- seq(0,1,length.out=101) # Values for relative connectivity

# Probability distribution assuming infinite settler pool and uniform prior
drc <- d.rel.conn.unif.prior(phi,p,k,n.obs)
prc <- p.rel.conn.unif.prior(phi,p,k,n.obs)
qrc <- q.rel.conn.unif.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval

# Test with finite settlement function and large (approx. infinite) settler pool
# Can be a bit slow for large settler pools
dis <- d.rel.conn.finite.settlement(0:(7*n.obs),p,k,n.obs,7*n.obs)

# Quantiles
qis <- q.rel.conn.finite.settlement(c(0.025,0.975),p,k,n.obs,7*n.obs)

# Finite settler pool
dfs <- d.rel.conn.finite.settlement(0:n.settlers,p,k,n.obs,n.settlers)

# Quantiles for the finite settler pool
qfs <- q.rel.conn.finite.settlement(c(0.025,0.975),p,k,n.obs,n.settlers)

# Make a plot of different distributions
plot(phi,drc,type="l",main="Probability of relative connectivity values",
     xlab=expression(phi),ylab="Probability density")
lines(phi,prc,col="blue")
lines((0:(7*n.obs))/(7*n.obs),dis*(7*n.obs),col="black",lty="dashed")
lines((0:n.settlers)/n.settlers,dfs*n.settlers,col="red",lty="dashed")
abline(v=qrc,col="black")
abline(v=qis/(7*n.obs),col="black",lty="dashed")
abline(v=qfs/n.settlers,col="red",lty="dashed")

Calculates unnormalized probability density for relative connectivity values from multiple distinct sites

Description

This functions calculates the unnormalized probability density function for the relative (to all settlers at the destination site) connectivity value for larval transport between multiple source sites to a destination site. An arbitrary number of source sites can be evaluated.

Usage

d.rel.conn.multinomial.unnorm(
  phis,
  ps,
  ks,
  n.sample,
  log = FALSE,
  dirichlet.prior.alphas = 1/(length(phis) + 1)
)

Arguments

Details

As this function returns the unnormalized probability density, it must be normalized somehow to be produce a true probability density. This can be acheived using a variety of approaches, including brute force integration of the unnormalized probability density and MCMC algorithms.

Value

The unnormalized probability density value. If log=TRUE, then the logarithm of the probability density value will be returned.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

Berger JO, Bernardo JM, Sun D (2015) Overall Objective Priors. Bayesian Analysis 10:189-221. doi:10.1214/14-BA915

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), dual.mark.transmission(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)

ps <- c(0.7,0.5) # Fraction of eggs "marked" at each source site
ks <- c(4,5) # Number of marked settlers among sample from each source site
n.sample <- 20 # Total sample size.  Must be >= sum(ks)


phis0 = runif(3,min=0.05)
phis0 = phis0 / sum(phis0)
phis0 = phis0[1:2] # Don't include relative connectivity of unknown sites

nbatch=1e4

library(mcmc)
ans = metrop(d.rel.conn.multinomial.unnorm,
             initial=phis0,nbatch=nbatch,scale=0.1,
             log=TRUE,ps=ps,ks=ks,n.sample=n.sample)
# A more serious test would adjust blen and scale to improve results, and would repeat
# multiple times to get results from multiple MCMC chains.

# Plot marginal distribution of relative connectivity from first site
h=hist(ans$batch[,1],xlab="Rel. Conn., Site 1",
       main="Relative Connectivity for Source Site 1")

# For comparison, add on curve that would correspond to single site calculation
phi = seq(0,1,length.out=40)
d1 = d.rel.conn.beta.prior(phi,ps[1],ks[1],n.sample)

lines(phi,d1*nbatch*diff(h$breaks)[1],col="red",lwd=5)

# Image plot of bivariate probability density
t=table(cut(ans$batch[,1],phi),cut(ans$batch[,2],phi))
image(t,col=heat.colors(12)[12:1],xlab="Rel. Conn., Site 1",ylab="Rel. Conn., Site 2")

# Add line indicate region above which one can never find results as that would 
# lead to a total connectivity great than 1
abline(1,-1,col="black",lty="dashed",lwd=3)

Functions for estimating the probability distribution of relative connectivity values as a weighted sum over possible input parameters

Description

These functions calculate the probability density function (d.rel.conn.multiple), the probability distribution function (aka the cumulative distribution function; p.rel.conn.multiple) and the quantile function (q.rel.conn.multiple) for the relative (to all settlers at the destination site) connectivity value for larval transport between a source and destination site. This version allows one to input multiple possible fractions of individuals (i.e., eggs) marked at the source site, multiple possible numbers of settlers collected and multiple possible marked individuals observed in the sample. This gives one the possibility to produce ensemble averages over different input parameter values with different probabilities of being correct.

Usage

d.rel.conn.multiple(
  phi,
  ps,
  ks,
  ns,
  weights = 1,
  d.rel.conn = d.rel.conn.beta.prior,
  ...
)

p.rel.conn.multiple(
  phi,
  ps,
  ks,
  ns,
  weights = 1,
  p.rel.conn = p.rel.conn.beta.prior,
  ...
)

q.rel.conn.multiple.func(
  ps,
  ks,
  ns,
  weights = 1,
  p.rel.conn = p.rel.conn.beta.prior,
  N = 1000,
  ...
)

q.rel.conn.multiple(
  q,
  ps,
  ks,
  ns,
  weights = 1,
  p.rel.conn = p.rel.conn.beta.prior,
  N = 1000,
  ...
)

Arguments

Details

If ps, ks, ns and weights can be scalars or vectors of the same length (or lengths divisible into that of the largest input parameter). weights are normalized to sum to 1 before being used to sum probabilities from each individual set of input parameters.

Value

Vector of probabilities or quantiles, or a function in the case of q.rel.conn.multiple.func

Functions

• d.rel.conn.multiple: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites for multiple possible p, k and n values.

• p.rel.conn.multiple: Estimates the cumulative probability distribution for relative connectivity between a paire of sites for multiple possible p, k and n values.

• q.rel.conn.multiple.func: Returns a function to estimate quantiles for the probability distribution function for relative connectivity between a pair of sites for multiple possible p, k and n values.

• q.rel.conn.multiple: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites for multiple possible p, k and n values.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multinomial.unnorm(), d.rel.conn.unif.prior(), dual.mark.transmission(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)

# p values have uniform probability between 0.1 and 0.4
p <- seq(0.1,0.8,length.out=100)

# Weights the same for all except first and last, which are halved
w <- rep(1,length(p))
w[1]<-0.5
w[length(w)]<-0.5

n <- 20 # Sample size
k <- 2 # Marked individuals in sample

# phi values to use for plotting distribution
phi <- seq(0,1,0.01)

prior.shape1 = 1 # Uniform prior
# prior.shape1 = 0.5 # Jeffreys prior

# Plot distribution
plot(phi,d.rel.conn.multiple(phi,p,k,n,w,prior.shape1=prior.shape1),
     main="Probability density for relative connectivity",
     xlab=expression(phi),
     ylab="Probability density",
     type="l")

# Add standard distributions for max and min p values
lines(phi,d.rel.conn.beta.prior(phi,min(p),k,n,prior.shape1=prior.shape1),
      col="red",lty="dashed")
lines(phi,d.rel.conn.beta.prior(phi,max(p),k,n,prior.shape1=prior.shape1),
      col="red",lty="dashed")

# Add some quantiles
q = q.rel.conn.multiple(c(0.025,0.25,0.5,0.75,0.975),
                        p,k,n,w,prior.shape1=prior.shape1)
abline(v=q,col="green")

Estimate the probability distribution of relative connectivity values assuming a uniform prior distribution

Description

These functions calculate the probability density function (d.rel.conn.unif.prior), the probability distribution function (aka the cumulative distribution function; p.rel.conn.unif.prior) and the quantile function (q.rel.conn.unif.prior) for the relative (to all settlers at the destination site) connectivity value for larval transport between a source and destination site given a known fraction of marked individuals (i.e., eggs) in the source population. A uniform prior is used for the relative connectivity value.

Usage

d.rel.conn.unif.prior(phi, p, k, n, log = FALSE, ...)

p.rel.conn.unif.prior(phi, p, k, n, log = FALSE, ...)

q.rel.conn.unif.prior(q, p, k, n, log = FALSE, ...)

Arguments

Details

Estimations of the probability distribution are derived from the Beta distribution (see dbeta) and should be exact to great precision.

Value

Vector of probabilities or quantiles.

Functions

• d.rel.conn.unif.prior: Returns the probability density for relative connectivity between a pair of sites

• p.rel.conn.unif.prior: Returns the cumulative probability distribution for relative connectivity between a paire of sites

• q.rel.conn.unif.prior: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multinomial.unnorm(), d.rel.conn.multiple(), dual.mark.transmission(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)

k <- 10 # Number of marked settlers among sample
n.obs <- 87 # Number of settlers in sample
n.settlers <- 100 # Total size of settler pool

p <- 0.4 # Fraction of eggs that was marked
phi <- seq(0,1,length.out=101) # Values for relative connectivity

# Probability distribution assuming infinite settler pool and uniform prior
drc <- d.rel.conn.unif.prior(phi,p,k,n.obs)
prc <- p.rel.conn.unif.prior(phi,p,k,n.obs)
qrc <- q.rel.conn.unif.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval

# Test with finite settlement function and large (approx. infinite) settler pool
# Can be a bit slow for large settler pools
dis <- d.rel.conn.finite.settlement(0:(7*n.obs),p,k,n.obs,7*n.obs)

# Quantiles
qis <- q.rel.conn.finite.settlement(c(0.025,0.975),p,k,n.obs,7*n.obs)

# Finite settler pool
dfs <- d.rel.conn.finite.settlement(0:n.settlers,p,k,n.obs,n.settlers)

# Quantiles for the finite settler pool
qfs <- q.rel.conn.finite.settlement(c(0.025,0.975),p,k,n.obs,n.settlers)

# Make a plot of different distributions
plot(phi,drc,type="l",main="Probability of relative connectivity values",
     xlab=expression(phi),ylab="Probability density")
lines(phi,prc,col="blue")
lines((0:(7*n.obs))/(7*n.obs),dis*(7*n.obs),col="black",lty="dashed")
lines((0:n.settlers)/n.settlers,dfs*n.settlers,col="red",lty="dashed")
abline(v=qrc,col="black")
abline(v=qis/(7*n.obs),col="black",lty="dashed")
abline(v=qfs/n.settlers,col="red",lty="dashed")

Sample LOD score data for simulated and real parent-child pairs

Description

This dataset contains both simulated and real 'log of the odds ratio' (LOD) scores for potential parent-child pairs of humbug damselfish (Dascyllus aruanus) from New Caledonia. Data was generated using FaMoz. In all cases, results are for the potential parent with the highest LOD score for a given larval fish (child). Simulated data is based on artificial children generated from either real potential parent-pairs (the 'in' group) or artificial parents generated from observed allelic frequencies (the 'out' group).

Format

A list with 3 elements:

in.group

5000 maximum LOD scores for simulated children from random crossing of real potential parents

out.group

5000 maximum LOD scores for simulated children from random crossing of artificial potential parents based on observed allelic frequencies

real.children

Maximum LOD scores for 200 real juvenile fish

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Gerber S, Chabrier P, Kremer A (2003) FAMOZ: a software for parentage analysis using dominant, codominant and uniparentally inherited markers. Molecular Ecology Notes 3:479-481. doi:10.1046/j.1471-8286.2003.00439.x

Kaplan et al. (submitted) Uncertainty in marine larval connectivity estimation

See Also

See also d.rel.conn.dists.func

Fraction of eggs marked for male and female mark transmission

Description

Estimates the fraction of eggs produced at the source site that are the result of crossing parents, one or both of which have been genotyped. Based on the assumption that probability of breeding between pairs of individuals is completely independent of whether or not one or more of those individuals was genotyped.

Usage

dual.mark.transmission(p.female, p.male = p.female)

Arguments

Value

A list with the following elements:

prob.matrix

2x2 matrix with probabilities for producing offspring with male or female known or unknown parents

p

fraction of all eggs produced at source site that will come from at least one genotyped parent

p.female.known

Fraction of eggs with a single known female parent among all eggs that have one or more known parents

p.male.known

Fraction of eggs with a single known male parent among all eggs that have one or more known parents

p.two.known.parents

Fraction of eggs with two known parents among all eggs that have one or more known parents

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multinomial.unnorm(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), optim.rel.conn.dists(), r.marked.egg.fraction()

Examples

library(ConnMatTools)
data(damselfish.lods)

# Histograms of simulated LODs
l <- seq(-1,30,0.5)
h.in <- hist(damselfish.lods$in.group,breaks=l)
h.out <- hist(damselfish.lods$out.group,breaks=l)

# PDFs for marked and unmarked individuals based on simulations
d.marked <- stepfun.hist(h.in)
d.unmarked <- stepfun.hist(h.out)

# Fraction of adults genotyped at source site
p.adults <- 0.25

# prior.shape1=1 # Uniform prior
prior.shape1=0.5 # Jeffreys prior

# Fraction of eggs from one or more genotyped parents
p <- dual.mark.transmission(p.adults)$p

# PDF for relative connectivity
D <- d.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Estimate most probable value for relative connectivity
phi.mx <- optim.rel.conn.dists(damselfish.lods$real.children,
                                    d.unmarked,d.marked,p)$phi

# Estimate 95% confidence interval for relative connectivity
Q <- q.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Plot it up
phi <- seq(0,1,0.001)
plot(phi,D(phi),type="l",
     xlim=c(0,0.1),
     main="PDF for relative connectivity",
     xlab=expression(phi),
     ylab="Probability density")

abline(v=phi.mx,col="green",lty="dashed")
abline(v=Q(c(0.025,0.975)),col="red",lty="dashed")

Compute some eigenvalues of a matrix

Description

This function computes a limited number of eigenvalues and eigenvectors of a matrix. It uses arpack function from the igraph package. If this package is not available, it will use the standard eigen function to do the calculation, but will issue a warning.

Usage

eigs(
  M,
  nev = min(dim(M)[1] - 1, 1),
  sym = sum(abs(M - t(M)))/sum(abs(M)) < 1e-10,
  which = "LM",
  use.arpack = TRUE,
  options.arpack = NULL
)

Arguments

Value

A list with at least the following two items:

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

See Also

See also arpack

Gamma distribution shape and scale parameters from mean and standard deviation, or vice-versa

Description

Calculates shape and scale parameters for a gamma distribution from the mean and standard deviation of the distribution, or vice-versa. One supplies either mean and sd or shape and scale and the function returns a list with all four parameter values.

Usage

gammaParamsConvert(...)

Arguments

Value

A list with mean, sd, shape and scale parameters of the corresponding gamma distribution.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

Examples

library(ConnMatTools)
mn <- 1
sd <- 0.4
l <- gammaParamsConvert(mean=mn,sd=sd)
x <- seq(0,2,length.out=50)

# Plot gamma and normal distributions - for sd << mean, the two should be very close
plot(x,dgamma(x,l$shape,scale=l$scale),
     main="Normal versus Gamma distributions",type="l")
lines(x,dnorm(x,l$mean,l$sd),col="red")

Hockey-stick settler-recruit relationship

Description

Calculates recruitment based on a settler-recruit relationship that increases linearly until it reaches a maximum values.

Usage

hockeyStick(S, slope = 1/0.35, Rmax = 1)

Arguments

Details

slope and Rmax can both either be scalars or vectors of the same length as S.

Value

A vector of recruitment values.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan, D. M., Botsford, L. W., and Jorgensen, S. 2006. Dispersal per recruit: An efficient method for assessing sustainability in marine reserve networks. Ecological Applications, 16: 2248-2263.

Uniform Laplacian connectivity matrix

Description

This function generates a connectivity matrix that is governed by a Laplacian distribution: D[i,j]=exp(abs(x[i]-y[i]-shift)/disp.dist)/2/disp.dist

Usage

laplacianConnMat(num.sites, disp.dist, shift = 0, boundaries = "nothing")

Arguments

Details

The boundary argument can have the following different values: "nothing" meaning do nothing special with boundaries; "conservative" meaning force columns of matrix to sum to 1; and "circular" meaning wrap edges.

Value

A square connectivity matrix

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan, D. M., Botsford, L. W., and Jorgensen, S. 2006. Dispersal per recruit: An efficient method for assessing sustainability in marine reserve networks. Ecological Applications, 16: 2248-2263.

See Also

See also DispersalPerRecruitModel

Examples

library(ConnMatTools)
cm <- laplacianConnMat(100,10,15,"circular")
image(cm)

Local retention of a connectivity matrix

Description

Local retention is defined as the diagonal elements of the connectivity matrix.

Usage

localRetention(conn.mat)

Arguments

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

Examples

library(ConnMatTools)
data(chile.loco)

sr <- selfRecruitment(chile.loco)
lr <- localRetention(chile.loco)
rlr <- relativeLocalRetention(chile.loco)

Merge subpopulations

Description

This function tries to merge random subopoulations, checking if the result is a better soluton to the minimization problem.

Usage

mergeSubpops(subpops.lst, conn.mat, beta)

Arguments

Value

List of the same format as subpops.lst, but with potentially fewer subpopulations.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

See Also

See also optimalSplitConnMat,

Maximum-likelihood estimate for relative connectivity given two distributions for scores for marked and unmarked individuals

Description

This function calculates the value for relative connectivity that best fits a set of observed score values, a pair of distributions for marked and unmarked individuals and an estimate of the fraction of eggs marked in the source population, p.

Usage

optim.rel.conn.dists(
  obs,
  d.unmarked,
  d.marked,
  p = 1,
  phi0 = 0.5,
  method = "Brent",
  lower = 0,
  upper = 1,
  ...
)

Arguments

Value

A list with results of optimization. Optimal fraction of marked individuals is in phi field. Negative log-likelihood is in the neg.log.prob field. See optim for other elements of list.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multinomial.unnorm(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), dual.mark.transmission(), r.marked.egg.fraction()

Examples

library(ConnMatTools)
data(damselfish.lods)

# Histograms of simulated LODs
l <- seq(-1,30,0.5)
h.in <- hist(damselfish.lods$in.group,breaks=l)
h.out <- hist(damselfish.lods$out.group,breaks=l)

# PDFs for marked and unmarked individuals based on simulations
d.marked <- stepfun.hist(h.in)
d.unmarked <- stepfun.hist(h.out)

# Fraction of adults genotyped at source site
p.adults <- 0.25

# prior.shape1=1 # Uniform prior
prior.shape1=0.5 # Jeffreys prior

# Fraction of eggs from one or more genotyped parents
p <- dual.mark.transmission(p.adults)$p

# PDF for relative connectivity
D <- d.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Estimate most probable value for relative connectivity
phi.mx <- optim.rel.conn.dists(damselfish.lods$real.children,
                                    d.unmarked,d.marked,p)$phi

# Estimate 95% confidence interval for relative connectivity
Q <- q.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Plot it up
phi <- seq(0,1,0.001)
plot(phi,D(phi),type="l",
     xlim=c(0,0.1),
     main="PDF for relative connectivity",
     xlab=expression(phi),
     ylab="Probability density")

abline(v=phi.mx,col="green",lty="dashed")
abline(v=Q(c(0.025,0.975)),col="red",lty="dashed")

Iteratively, optimally split a connectivity matrix

Description

Algorithm for iteratively determining subpopulations of highly-connected sites. Uses an iterative method described in Jacobi et al. (2012)

Usage

optimalSplitConnMat(
  conn.mat,
  normalize.cols = TRUE,
  make.symmetric = "mean",
  remove.diagonal = FALSE,
  cycles = 2,
  betas = betasVectorDefault(ifelse(normalize.cols, dim(conn.mat)[2],
    prod(dim(conn.mat))/sum(conn.mat)), steps),
  steps = 10,
  ...
)

Arguments

Value

A list with the following elements:

Note

In Jacobi et al. (2012) paper, the connectivity matrix is oriented so that C_ij is dispersal from i to j, whereas in this R package, the connectivity matrix is oriented so that C_ij is dispersal from j to i. This choice of orientation is arbitrary, but one must always be consistent. From j to i is more common in population dynamics because it works well with matrix multiplication (e.g., settlers = conn.mat %*% eggs).

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

See Also

See also splitConnMat, recSplitConnMat, mergeSubpops, qualitySubpops

Examples

library(ConnMatTools)
data(chile.loco)

num <- prod(dim(chile.loco)) / sum(chile.loco)
betas <- betasVectorDefault(n=num,steps=4)
chile.loco.split <- optimalSplitConnMat(chile.loco,normalize.cols=FALSE,
                                        betas=betas)

# Extra 3rd division
print(paste("Examining split with",names(chile.loco.split$best.splits)[1],
            "subpopulations."))
pops <- subpopsVectorToList(chile.loco.split$subpops[,chile.loco.split$best.splits[[1]]$index])

reduce.loco <- reducedConnMat(pops,chile.loco)

sr <- selfRecruitment(reduce.loco)
lr <- localRetention(reduce.loco)
rlr <- relativeLocalRetention(reduce.loco)

Returns probability a set of observations correspond to marked individuals

Description

This function returns the probability each of a set of observations corresponds to a marked individual given the distribution of scores for unmarked and marked individuals and the fraction of individuals that are marked.

Usage

prob.marked(obs, d.unmarked, d.marked, phi = 0.5, p = 1)

Arguments

Value

A vector of the same size as obs containing the probability that each individual is marked

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

See also d.rel.conn.dists.func, optim.rel.conn.dists.

Function to select optimal network of protected areas based on connectivity

Description

This function finds the optimal network of protected areas based on connectivity using the eigenvalue perturbation approach described in Nilsson Jacobi & Jonsson (2011).

Usage

protectedAreaSelection(
  conn.mat,
  nev = dim(conn.mat)[1],
  delta = 0.1,
  theta = 0.05,
  M = 20,
  epsilon.lambda = 1e-04,
  epsilon.uv = 0.05,
  only.list = T,
  ...
)

Arguments

Value

If only.list is TRUE, just returns the list of selected sites. If FALSE, then result will be a list containing selected sites and predicted impact of each selected site on the eigenvalues.

Author(s)

Marco Andrello marco.andrello@gmail.com

References

Jacobi, M. N., and Jonsson, P. R. 2011. Optimal networks of nature reserves can be found through eigenvalue perturbation theory of the connectivity matrix. Ecological Applications, 21: 1861-1870.

Quality measure for subpopulation division

Description

A measure of the leakage between subpopulations for a given division of the connectivity matrix into subpopulations. This statistic is equal to 1 - mean(RLR) of the reduced connectivity matrix, where RLR=relative local retention (relativeLocalRetention), i.e., the fraction of settling individuals that originated at their site of settlement.

Usage

qualitySubpops(subpops.lst, conn.mat)

Arguments

Value

The quality statistic.

A smaller value of the quality statistic indicates less leakage.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

See Also

See also optimalSplitConnMat, subpopsVectorToList, relativeLocalRetention

Estimates of fraction of eggs marked accounting for variability in reproductive output

Description

This function estimates the fraction of eggs "marked" at a site (where the "mark" could be micro-chemical or genetic) taking into account uncertainty in female (and potentially male in the case of dual genetic mark transmission) reproductive output. It generates a set of potential values for the fraction of eggs marked assuming that reproductive output of each marked or unmarked mature individual is given by a random variable drawn from a single probability distribution with known mean and standard deviation (or equivalently coefficient of variation) and that the numbers of marked and unmarked individuals are large enough that the central limit theorem applies and, therefore, their collective reproductive outputs are reasonably well described by a gamma distribution whose mean and standard deviation are appropriately scaled based on the number of individual reproducers. The function also returns the total egg production corresponding to each fraction of marked eggs, needed for estimating absolute connectivity values (i.e., elements of the connectivity matrix needed for assessing population persistence).

Usage

r.marked.egg.fraction(
  n,
  n.females,
  n.marked.females = round(n.females * p.marked.females),
  mean.female = 1,
  cv.female,
  dual = FALSE,
  male.uncert = FALSE,
  n.males = n.females,
  n.marked.males = tryCatch(round(n.males * p.marked.males), error = function(e)
    n.marked.females),
  mean.male = mean.female,
  cv.male = cv.female,
  p.marked.females,
  p.marked.males = p.marked.females
)

Arguments

Value

A list with the following elements:

p

Vector of length n with estimates for fraction of marked eggs

eggs

Vector of length n with estimates for total egg production

marked.eggs

Vector of length n with estimates for total number of marked eggs produced

sperm

Only returned if dual=TRUE. If male.uncert=FALSE, then a scalar equal to n.males. Otherwise, a vector of length n with estimates for total sperm production

marked.sperm

Only returned if dual=TRUE. If male.uncert=FALSE, then a scalar equal to n.marked.males. Otherwise, a vector of length n with estimates for total marked sperm production

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

See Also

Other connectivity estimation: d.rel.conn.beta.prior(), d.rel.conn.dists.func(), d.rel.conn.finite.settlement(), d.rel.conn.multinomial.unnorm(), d.rel.conn.multiple(), d.rel.conn.unif.prior(), dual.mark.transmission(), optim.rel.conn.dists()

Examples

library(ConnMatTools)

n.females <- 500
n.marked.females <- 100
p.marked.females <- n.marked.females/n.females
mn <- 1
cv <- 1
# Numbers of males and marked males and variance in male sperm production
# assumed the same as values for females

# Random values from distribution of pure female mark transmission
F=r.marked.egg.fraction(1000,n.females=n.females,n.marked.females=n.marked.females,
                        mean.female=mn,cv.female=cv)

# Random values from distribution of dual female-male mark transmission, but
# fraction of marked eggs only depends on fraction of marked males
Fm=r.marked.egg.fraction(1000,n.females=n.females,n.marked.females=n.marked.females,
                        mean.female=mn,cv.female=cv,dual=TRUE,male.uncert=FALSE)

# Random values from distribution of dual female-male mark transmission, with
# fraction of marked eggs depending on absolute marked and unmarked sperm output
FM=r.marked.egg.fraction(1000,n.females=n.females,n.marked.females=n.marked.females,
                         mean.female=mn,cv.female=cv,dual=TRUE,male.uncert=TRUE)

# Plot of pure female mark transmission
hist(F$p,50,main="Female mark transmission",
     xlab="Fraction of marked eggs",
     ylab="Frequency")

# Female+male mark transmission, but no variability in male mark transmission
h <- hist(Fm$p,50,main="Female+male mark transmission, no male uncert.",
          xlab="Fraction of marked eggs",
          ylab="Frequency")
hh <- hist((1-p.marked.females)*F$p + p.marked.females,
           breaks=c(-Inf,h$breaks,Inf),plot=FALSE)
lines(hh$mids,hh$counts,col="red")

# Plot of pure female mark transmission
h <- hist(FM$p,50,plot=FALSE)
hh <- hist(Fm$p,
           breaks=c(-Inf,h$breaks,Inf),plot=FALSE)
plot(h,ylim=c(0,1.1*max(hh$counts,h$counts)),
     main="Female+Male mark transmission, male uncert.",
     xlab="Fraction of marked eggs",
     ylab="Frequency")
lines(hh$mids,hh$counts,col="red")

Recursively subdivides a set of subpoplations

Description

This funtion recursively splits each subpopulation of a list of subpopulations until none of the subpopulations can be split further to improve the minimization.

Usage

recSplitConnMat(subpops.lst, conn.mat, beta, ...)

Arguments

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

See Also

See also optimalSplitConnMat, splitConnMat, subpopsVectorToList

Reduced connectivity matrix according to a set of subpopulations

Description

Reduces a connectivity matrix based on a set of subpopulations. If there are N subpopulations, then the reduced matrix will have dimensions NxN. The reduced matrix will be ordered according to the order of subpopulations in subpops.lst.

Usage

reducedConnMat(subpops.lst, conn.mat)

Arguments

Value

A reduced connectivity matrix. The sum of all elements of this reduced connectivity matrix will be equal to the sum of all elements of the original connectivity matrix.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

See Also

See also qualitySubpops

Examples

library(ConnMatTools)
data(chile.loco)

num <- prod(dim(chile.loco)) / sum(chile.loco)
betas <- betasVectorDefault(n=num,steps=4)
chile.loco.split <- optimalSplitConnMat(chile.loco,normalize.cols=FALSE,
                                        betas=betas)

# Extra 3rd division
print(paste("Examining split with",names(chile.loco.split$best.splits)[1],
            "subpopulations."))
pops <- subpopsVectorToList(chile.loco.split$subpops[,chile.loco.split$best.splits[[1]]$index])

reduce.loco <- reducedConnMat(pops,chile.loco)

sr <- selfRecruitment(reduce.loco)
lr <- localRetention(reduce.loco)
rlr <- relativeLocalRetention(reduce.loco)

Relative local retention of a connectivity matrix

Description

Relative local retention is defined as the diagonal elements of the connectivity matrix divided by the sum of the corresponding column of the connectivity matrix.

Usage

relativeLocalRetention(conn.mat)

Arguments

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

Examples

library(ConnMatTools)
data(chile.loco)

sr <- selfRecruitment(chile.loco)
lr <- localRetention(chile.loco)
rlr <- relativeLocalRetention(chile.loco)

Self recruitment of a connectivity matrix

Description

If egg production is uniform over sites, then self recruitment is defined as the diagonal elements of the connectivity matrix divided by the sum of the corresponding row of the connectivity matrix. If not, then the elements of the dispersal matrix must be weighted by the number of eggs produced.

Usage

selfRecruitment(conn.mat, eggs = NULL)

Arguments

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

Examples

library(ConnMatTools)
data(chile.loco)

sr <- selfRecruitment(chile.loco)
lr <- localRetention(chile.loco)
rlr <- relativeLocalRetention(chile.loco)

Correction for slope of settler-recruit relationship

Description

This function corrects the slope of the settler-recruit curve so that the collapse point of the spatially-explicit population model corresponding to the connectivity matrix agrees with that of the global non-spatially-explicit model. Uses the method in White (2010).

Usage

settlerRecruitSlopeCorrection(
  conn.mat,
  slope = 1,
  natural.LEP = 1,
  critical.FLEP = 0.35,
  use.arpack = TRUE
)

Arguments

Value

The slope argument corrected so that collapse happens when LEP is critical.FLEP * natural.LEP.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

White, J. W. 2010. Adapting the steepness parameter from stock-recruit curves for use in spatially explicit models. Fisheries Research, 102: 330-334.

See Also

See also eigs, arpack

Split connectivity matrix into subpopulations

Description

This function tries to optimally split a given subpopulation into two smaller subpopulations.

Usage

splitConnMat(
  indices,
  conn.mat,
  beta,
  tries = 5,
  threshold = 1e-10,
  alpha = 0.1,
  maxit = 500
)

Arguments

Value

List with one or two elements, each containing a vector of indices of sites in a subpopulations

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

References

Jacobi, M. N., Andre, C., Doos, K., and Jonsson, P. R. 2012. Identification of subpopulations from connectivity matrices. Ecography, 35: 1004-1016.

See Also

See also optimalSplitConnMat, recSplitConnMat, subpopsVectorToList

Create a probability density step function from a histogram object

Description

This function creates a step function from the bars in a histogram object. By default, the step function will be normalized so that it integrates to 1.

Usage

stepfun.hist(h, ..., normalize = TRUE)

Arguments

Value

A function of class stepfun. The height of the steps will be divided by the distance between breaks and possibly the total count.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

See Also

See also d.rel.conn.dists.func, optim.rel.conn.dists.

Examples

library(ConnMatTools)
data(damselfish.lods)

# Histograms of simulated LODs
l <- seq(-1,30,0.5)
h.in <- hist(damselfish.lods$in.group,breaks=l)
h.out <- hist(damselfish.lods$out.group,breaks=l)

# PDFs for marked and unmarked individuals based on simulations
d.marked <- stepfun.hist(h.in)
d.unmarked <- stepfun.hist(h.out)

# Fraction of adults genotyped at source site
p.adults <- 0.25

# prior.shape1=1 # Uniform prior
prior.shape1=0.5 # Jeffreys prior

# Fraction of eggs from one or more genotyped parents
p <- dual.mark.transmission(p.adults)$p

# PDF for relative connectivity
D <- d.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Estimate most probable value for relative connectivity
phi.mx <- optim.rel.conn.dists(damselfish.lods$real.children,
                                    d.unmarked,d.marked,p)$phi

# Estimate 95% confidence interval for relative connectivity
Q <- q.rel.conn.dists.func(damselfish.lods$real.children,
                           d.unmarked,d.marked,p,
                           prior.shape1=prior.shape1)

# Plot it up
phi <- seq(0,1,0.001)
plot(phi,D(phi),type="l",
     xlim=c(0,0.1),
     main="PDF for relative connectivity",
     xlab=expression(phi),
     ylab="Probability density")

abline(v=phi.mx,col="green",lty="dashed")
abline(v=Q(c(0.025,0.975)),col="red",lty="dashed")

Convert subpopulation vector to a list of indices

Description

A helper function to convert a vector of subpopulation identifications into a list appropriate for recSplitConnMat, qualitySubpops, etc.

Usage

subpopsVectorToList(x)

Arguments

Details

Note that subpopulations list will be ordered according to the numerical order of the subpopulation indices in the original matrix, which will not necessarily be that of the spatial order of sites in the original connectivity matrix.

Value

A list where each element is a vector of indices for a given subpopulation.

Author(s)

David M. Kaplan dmkaplan2000@gmail.com

See Also

See also recSplitConnMat, qualitySubpops