powh.RdEstimates growth and mortality parameters from length frequency data.
# S4 method for numeric,numeric powh(len, n, weights = FALSE, fromMode = FALSE, linf = 0)
| len | vector with length distribution |
|---|---|
| n | vector with numbers in each length bin |
| weights | weights for observations |
| fromMode | booelean set to FALSE by default |
| linf | allows linf to be fixed if >0 |
| ... | any other argument, i.e. weights =TRUE, fromMode =FALSE |
a data.frame with mn (mean), diff (difference),
len (length) and n (frequency)
Beverton and Holt (1956) developed a method to estimate population parameters such as total mortality (Z) from length data i.e.
$$Z=K\frac{L_{infinity}-\overline{L}}{\overline{L}-L'}$$
Powell (1979) then developed a method, extended by Wetherall et al. (1987), to estimate growth and mortality parameters. This assumes that the right hand tail of a length frequency distribution is determined by the asymptotic length L and the ratio between Z and the growth rate K.
The Beverton and Holt methods assumes good estimates for K and $L_infinity$, while the Powell-Wetherall method only requires an estimate of K, since $L_infinity$ is estimated by the method as well as Z/K. These method therefore provide estimates for Z/K, if K is unknown and Z if K is known.
As well as assuming that growth follows the von Bertalanffy growth function, it is also assumed that the population is in a steady state with constant exponential mortality, no changes in selection pattern of the fishery and constant recruitment.
In the Powell-Wetherall method L' can take any value between the smallest and largest sizes. Equation 1 then provides a series of estimates of Z and since
$$ \overline{L}-L'=a+bL' $$
a and b can be estimated by a regression analysis where
$$b={-K}/{Z+K}$$ $$a=-bL_{infinity}$$
Therefore plotting $$\overline{L}-L$$ against $L'$ provides an estimate of $L_infinity$ and Z/K from
$$L_{infinity}=-a/b$$ $$Z/K={-1-b}/{b}$$
If K is known Z can also be esimated
R. Beverton and S. Holt. Review of method for estimating mortality rates in exploited fish populations, with special reference to sources of bias in catch sampling. Rapports et Proces-Verbaux., 140(1): 67--83, 1956.
D. G. Powell. Estimation of mortality and growth parameters from the length frequency of a catch [model]. Rapports et Proces-Verbaux des Reunions, 175, 1979.
J. Wetherall, J. Polovina, and S. Ralston. Estimating growth and mortality in steady-state fish stocks from length-frequency data. ICLARM Conf. Proc, pages 53--74, 1987.
# NOT RUN { data(cas) pw=ddply(subset(cas), .(year), function(cas) powh(cas$len,cas$n)$data) pw=transform(pw, lustrum=(year%/%5)*5, yr =year-(year%/%5)*5, weight=ifelse(len>=100&len<=200,1,0)) ggplot(pw)+ geom_line(aes(len,diff,colour=factor(yr),group=year))+ scale_x_continuous(limits=c(0,300)) + xlab("Length (cm)")+ ylab("Difference between Length and Mean Size")+ geom_smooth(aes(len,diff,weight=weight), method="lm",col="red",size=1.25,alpha=.1)+ theme_bw()+theme(legend.position="none") # }