Production models with time series fitting
Description
This function applies the production models under nonequilibrium conditions by applying time series fitting using nonlinear least squares minimisation.
Usage
1 2 
Arguments
data 
a dataframe of parameters

method 
indicating if Schaefer or Fox model should be applied. First assumes a logistic relationship between growth rate and biomass, whereas second assumes it to foolow the Gompertz distribution (Richards 1959). Default is the dynamic Schaefer model. 
B0_init 
numeric; if realistic initial estimate for virgin biomass is available.
If NA initial estimate for virgin biomass is set to two times average yield of all
or part of yield values (see 
B0_est 
intital value of virgin biomass estimating using all yield values (NA) or first years of time series, then provide numerical representing number of years 
effort_unit 
multiplication factor for the unit of effort. Default is 1. 
plot 
logical; if TRUE (default) a graph is displayed 
Details
Either catch per unit of effort (CPUE) is inserted
into the model directly (by a column CPUE
) or CPUE is calculated from
the catch and effort, then these two vectors should have required units.
Whenever a good estimate for the virigin biomass is available, this estimate
should be inserted for B_init
. The default approach for the initial
estimate of the virgin biomass is to multiply the average yield by 2 (Dharmendra
and Solmundsson, 2005). Alternatively, just a part of the time series of
yield values can be choosen to represent the virgin biomass.
The minimisation procedure is based on least error sum of squares (SSE). For
the logistic (Schaefer) method the standard calculation of SSE is applied
(sum((CPUE  predicted CPUE)^2)
), for
the method with Gompertz distribution (Fox) SSE is calculated according to
the Thiel's U statistic sqrt(sum(CPUE  predicted CPUE)/sum(CPUE(t)  CPUE(t1)))
(Wittink, 1988).
Value
A list with the input parameters and following list objects:

Bvec: biomass vector,

CPUE_hat: predicted CPUE,

K: carrying capacity,

r: population growth rate,

q: catchability coefficient,

MSY: maximum sustainabale yield (MSY),

Bmsy: biomass at MSY,

Emsy: fishing effort at MSY

Fmsy: fishing mortality at MSY,
References
Dharmendra, D., Solmundsson, J., 2005. Stock assessment of the offshore Mauritian banks using dynamic biomass models and analysis of length frequency of the Sky Emperor (Lethrinus mahsena). Fisheries Training Program The United Nations University, 61
Hilborn, R. and Walters, C., 1992. Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty. Chapman and Hall, New York
Prager, M. H., 1994. A suite of extensions to a nonequilibrium surplus production model. Fishery Bulletin 92: 374389
Richards, F. J., 1959. A flexible growth function for empirical use. Journal of experimental Botany, 10(2), 290301.
Wittink, D. R., 1988. The application of regression analysis. Allyn and Bacon. Inc. Boston. MA. 324p.
Examples
1 2 3  data(emperor)
prod_mod_ts(emperor, method = "Schaefer")
prod_mod_ts(emperor, method = "Fox")
