mdd.RdLorenzen natural mortality relationship where M is a function of weight, modified to explicitly included M as a function of numbers in a cohort, i.e. density dependence
# S4 method for FLQuant,FLPar
mdd(object, params, scale, k = 1, m = gislason)mass at which M is to be predicted
an FLPar with two values; i.e. a equal to M at unit mass and b a power term; defaults are a=0.3 and b=-0.288
reference
rate of change in density dependence
function with mortality model, by default gisalson
other arguments, such as scale, e.g. stock numbers now relative to a reference level, e.g. at virgin biomass and k steepness of relationship
@details
The Lorenzen natural mortality relationship is a function of mass-at-age i.e. M=a*wt^b
The relationship can be explained by population density, since as fish grow they also die and so there is potentially less competition for resources between larger and older fish. Density dependence can be modelled by a logistic function, a sigmoid curve (or S shaped) curve, with equation
f(x)=L/(1+exp(-k(x-x0)))
where e is the natural logarithm base (also known as Euler's number), x0 is the x-value of the sigmoid's midpoint, L is the curve's maximum value, and k the steepness of the curve.
Combining the two functions gives
M=aL/(1+exp(-k(n-ref)))*wt^b;
if (FALSE) {
library(FLBRP)
library(FLife)
data(teleost)
par=teleost[,"Hucho hucho"]
par=lhPar(par)
hutchen=lhEql(par)
scale=stock.n(hutchen)[,25]%*%stock.wt(hutchen)
scale=(stock.n(hutchen)%*%stock.wt(hutchen)%-%scale)%/%scale
m=mdd(wt2len(stock.wt(hutchen),par),params=par,scale,k=.9)
ggplot(as.data.frame(m))+
geom_line(aes(age,data,col=factor(year)))+
theme(legend.position="none")+
scale_x_continuous(limits=c(0,15))
m=mdd(stock.wt(hutchen),params=FLPar(m1=3,m2=-0.288),scale,k=1.2,m=lorenzen)
library(FLife)
}