A range of stock-recruitment (SR) models commonly used in fisheries science are provided in FLCore.
Usage
ricker()
bevholt()
bevholtDa()
bevholtss3()
segreg()
geomean()
shepherd()
cushing()
rickerSV()
bevholtSV()
shepherdSV()
bevholtAR1()
rickerAR1()
segregAR1()
rickerCa()
survRec(ssf, R0, Sfrac, beta, SF0 = ssf[, 1])
bevholtsig()
mixedsrr()Details
Each method is defined as a function returning a list with one or more elements as follows:
model: Formula for the model, using the slot names rec and ssb to refer to the usual inputs
logl: Function to calculate the loglikelihood of the given model when estimated through MLE (See
fmle)initial: Function to provide initial values for all parameters to the minimization algorithms called by
fmleornls. If required, this function also has two attributes,lowerandupper, that give lower and upper limits for the parameter values, respectively. This is used by some of the methods defined inoptim, like"L-BFGS-B".
The model<- method for FLModel can then be called
with value being a list as described above, the name of the function
returning such a list, or the function itself. See the examples below.
Several functions to fit commonly-used SR models are available. They all use
maximum likelihood to estimate the parameters through the method
loglAR1.
ricker: Ricker stock-recruitment model fit: $$R = a S e^{-b S}$$ a is related to productivity (recruits per stock unit at small stock size) and b to density dependence. (a, b > 0).
bevholt: Beverton-Holt stock-recruitment model fit: $$R = \frac{a S}{b + S}$$ a is the maximum recruitment (asymptotically) and b is the stock level needed to produce the half of maximum recruitment \(\frac{a}{2}\). (a, b > 0).
segreg: Segmented regression stock-recruitment model fit: $$R = \mathbf{ifelse}(S \leq b, a S, a b)$$ a is the slope of the recruitment for stock levels below b, and \(a b\) is the mean recruitment for stock levels above b. (a, b > 0).
geomean: Constant recruitment model fit, equal to the historical geometric mean recruitment. $$(R_1 R_2 \ldots R_n)^{1/n} = e^{\mathbf{mean}(\log(R_1),\ldots , }$$$$ \log(R_n))$$
shepherd: Shepherd stock-recruitment model fit: $$R = \frac{a S}{1+(\frac{S}{b})^c}$$ a represents density-independent survival (similar to a in the Ricker stock-recruit model), b the stock size above which density-dependent processes predominate over density-independent ones (also referred to as the threshold stock size), and c the degree of compensation.
cushing: Cushing stock-recruitment model fit: $$R = a S e^{b}$$ This model has been used less often, and is limited by the fact that it is unbounded for b>=1 as S increases. (a, b > 0).
Stock recruitment models parameterized for steepness and virgin biomass:
rickerSV: Fits a ricker stock-recruitment model parameterized for steepness and virgin biomass. $$a = e^{\frac{b \cdot vbiomass}{spr0}}$$ $$b = \frac{\log(5 \cdot steepness)}{0.8 \cdot vbiomass}$$
bevholtSV: Fits a Beverton-Holt stock-recruitment model parameterised for steepness and virgin biomass. $$a = \frac{4 \cdot vbiomass \cdot steepness}{(spr0 \cdot (5 \cdot steepness-1.0}$$ $$b = \frac{vbiomass (1.0-steepness)}{5 \cdot steepnes-1.0}$$
sheperdSV: Fits a shepher stock-recruitment model parameterized for steepness and virgin biomass. $$a = \frac{1.0+(\frac{vbiomass}{b})^c}{spr0}$$ $$b = vbiomass (\frac{0.2-steepness}{steepness (0.2)^c - 0.2})^ (\frac{-1.0}{c})$$
Models fitted using autoregressive residuals of first order:
bevholtAR1, rickerAR1, segregAR1: Beverton-Holt, Ricker and segmented regression stock-recruitment models with autoregressive normal log residuals of first order. In the model fit, the corresponding stock-recruit model is combined with an autoregressive normal log likelihood of first order for the residuals. If \(R_t\) is the observed recruitment and \(\hat{R}_t\) is the predicted recruitment, an autoregressive model of first order is fitted to the log-residuals, \(x_t = \log(\frac{R_t}{\hat{R}_t})\). $$x_t=\rho x_{t-1} + e$$ where \(e\) follows a normal distribution with mean 0: \(e \sim N(0, \sigma^2_{AR})\).
Ricker model with one covariate. The covariate can be used, for example, to account for an enviromental factor that influences the recruitment dynamics. In the equations, c is the shape parameter and X is the covariate.
rickerCa: Ricker stock-recruitment model with one multiplicative covariate. $$R = a (1- c X) S e^{-b S}$$
References
Beverton, R.J.H. and Holt, S.J. (1957) On the dynamics of exploited fish populations. MAFF Fish. Invest., Ser: II 19, 533.
Needle, C.L. Recruitment models: diagnosis and prognosis. Reviews in Fish Biology and Fisheries 11: 95-111, 2002.
Ricker, W.E. (1954) Stock and recruitment. J. Fish. Res. Bd Can. 11, 559-623.
Shepherd, J.G. (1982) A versatile new stock-recruitment relationship for fisheries and the construction of sustainable yield curves. J. Cons. Int. Explor. Mer 40, 67-75.
Examples
# inspect the output of one of the model functions
bevholt()
#> $logl
#> function(a, b, rec, ssb)
#> loglAR1(log(rec), log(a*ssb/(b+ssb)))
#> <bytecode: 0x55fb4b8f4808>
#> <environment: 0x55fb54ee5000>
#>
#> $model
#> rec ~ a * ssb/(b + ssb)
#> <environment: 0x55fb54ee5000>
#>
#> $initial
#> function(rec, ssb) {
#> a <- max(quantile(c(rec), 0.75, na.rm = TRUE))
#> b <- max(quantile(c(rec)/c(ssb), 0.9, na.rm = TRUE))
#> return(FLPar(a = a, b = a/b))}
#> <bytecode: 0x55fb4b8fb8e0>
#> <environment: 0x55fb54ee5000>
#> attr(,"lower")
#> [1] -Inf -Inf
#> attr(,"upper")
#> [1] Inf Inf
#>
names(bevholt())
#> [1] "logl" "model" "initial"
bevholt()$logl
#> function(a, b, rec, ssb)
#> loglAR1(log(rec), log(a*ssb/(b+ssb)))
#> <bytecode: 0x55fb4b8f4808>
#> <environment: 0x55fb54f54f88>
# once an FLSR model is in the workspace ...
data(nsher)
# the three model-definition slots can be modified
# at once by calling 'model<-' with
# (1) a list
model(nsher) <- bevholt()
# (2) the name of the function returning this list
model(nsher) <- 'bevholt'
# or (3) the function itself that returns this list
model(nsher) <- bevholt