[R] model of fish over exploitation

[bernard]

Dear all,

I have a serious problem to solve my model. I study over exploitation of
fish in the bay of biscay (france). I know only the level of catch and
the fishing effort (see data below) by year.

My model is composed by the following equations:

* the growth function
Gt(St) = r*St*(1-St/sbar)
with Gt the growth of each period t
r intrinsec growth of the stock
sbar carriyng capacity of the stock
St the fish stock of each period t

* the production function
Ht(St, Xt) = alpha*St*Xt
Ht the catch for each period t
Xt fishing effort for each period t
alpha parameter of boat productivity

* the dynamic of the fish stock
S(t+1) = S(t) + Gt - Ht


I would like to modelise the following system:

S_(t+1) = S_t + G_t - H_t
G_t = r*S_t*(1-S_t / sbar)
H_t = alpha * S_t * X_t
S_1961 = S_0

I know only H_t on period (1961 - 1994) and X_t on the same period.
I don't know r, sbar, alpha and S_0 (the initial level of the stock)
(and of course S_t on this period) and I want to estimate this four
parameters.

I have written something like that:

*******************************************
library(nls)
data <- read.table("fish.dat", header=TRUE)
attach(data)
names(data)

fct <- function(x, param)
{
  r     <- param[1]
  sbar  <- param[2]
  alpha <- param[3]
  s0    <- param[4]

for (i in 1:length(year))
  g[i] <- r*S[i]*(1-s[i] / sbar)
  h[i] <- alpha * s[i] * x[i]
  s[i+1] <- s[i] + g[i] - h[i]
}

mod <- nls (catch ~ fct(effort, catch, c(r, sbar, alpha, s0)), start =
c(r = 1.0, sbar = 100.0, alpha = 0.5, S0 = 70.0))

******************************************* 

... but, of course, it doesn't work !!

Could you help me please and could you answerd in my personnal email ? I
have put at the end data of my problem.

best regards to all of you

Pascal BERNARD

----------------------------------------
Researcher
LEN-CORRAIL
University of Nantes
Faculty of Economy
Chemin de la Censive du Tertre
44322 Nantes cedex 03
France
pascal.bernard at sc-eco.univ-nantes.fr
----------------------------------------

fish.dat:
************************
year catch effort
1961 41.400 45.03512 
1962 51.800 50.05673 
1963 44.300 44.30000 
1964 48.000 44.54000 
1965 44.826 59.97878 
1966 39.208 45.37687 
1967 48.278 46.60833 
1968 37.819 52.24526 
1969 31.992 54.11967 
1970 29.894 35.60816 
1971 39.406 61.24754 
1972 34.279 54.76161 
1973 27.958 46.56643 
1974 36.407 28.51477 
1975 27.827 27.16532 
1976 33.710 38.83327 
1977 32.888 22.07106 
1978 35.804 31.36205 
1979 38.950 25.68735 
1980 29.157 19.38004 
1981 23.748 21.78884 
1982 28.333 20.10470 
1983 31.945 27.18085 
1984 18.434 17.92367 
1985 22.531 18.97028 
1986 25.587 22.37778 
1987 29.777 16.89844 
1988 27.906 20.19613 
1989 25.757 16.42839 
1990 24.503 15.57284 
1991 16.608 17.14402 
1992 18.162 15.78574 
1993 18.371 12.12064 
1994 16.993 10.31185