[R] model of fish over exploitation

[(Ted Harding)]

On 02-Dec-03 bernard wrote:
> 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 don't know (though no doubt others do) about fitting non-linear
dynamic models to time-series data in R. However, I can suggest
the following initial approach, which while rather crude may
be useful.

Let R[t] = H[t]/X[t] be the CPUE (catch per unit effort). Proceeding 
for the moment as though your equations were exactly deterministic,
you get

  R[t] = alpha*S[t]

from which S[t] = R[t]/alpha, and then

  R[t+1] = R[t] + r*R[t]*(1 - R[t]/(alpha*sbar)) - alpha*H[t]

which is an equation which involves only variables which can be
evaluated from your data, and the parameters r, alpha and sbar.

The real-life deviation of data from model can be represented
in this by adding a stochastic term.

As a first approximation, consider linearising this equation
by ignoring the non-linear term:

  R[t+1] = R[t] + r*R[t] - alpha*H[t]
         = R[t](1+r) - alpha*H[t]

(this is tantamount to assuming that the fish stock S[t] is well
 below the "carrying capacity" which, for a thoroughly expoited
 fishery in the Bay of Biscay, may be a realistic assumption ... ).

You are now in the domain of linear AR time series with a known
driving term, and I'm sure people can point you to good procedures
in R for estimating the paranetrs (1+r) and alpha in this.

At this stage, what you have obtained only has the status of an
initial indication of realistic values, since it is based on a
somewhat brutal manipulation of your original model.

However, you can now go back to your original equations. Your
most important task is to think of realistic statistical models
for the difference between your data and your equations, i.e. how
you will incorporate the stochastic errors. I would suggest here
that (based on experience with fisheries data) you work with
the logarithms of the variables, and make the errors additive
with respect to these; and "normally distributed" will probably
be an assumption which you can't disprove ...

Now, with a given hypothetical S[0] (actually I think you should
call this S[1] if you mean the Stock in 1961), you can explore the
likelihood function for the parameters. Find the maximum by some
method. Now you can vary S[0], and similarly get the maximising
value of this. A feasible starting value for S[0] could be
what is implied by S[0] = H[0]/(alpha*X[0]).

You may well find that r and alpha do not differ much from the
values you got in the first round. Your only information about
sbar will come from the non-linear term, whose coefficient is
likely to be poorly estimated (I'm guessing here ... ).

Incidentally, if you look at a plot of log(CPUE) against year:

  plot(year,log(catch)-log(effort))

it seems that something interesting happened around 1973-1975
(or, perhaps, something interesting was going on prior to this
period), since CPUE makes a sudden jump (it's not clear whether
1961-1967 is similar to 1975+ or whether the whole series is on
the low side prior to 1974; but there is at least a series of
low values from 1968-1973). Are these due to low stocks?
To mismatch between where people fished and where the fish were?
To competition from other fishing fleets? To non-declaration
of catches? ... ? Whatever the reason, this range of years is
going to complicate your model fitting!

I hoped this helps, and apologies for not knowing enough about
what's available on this front in R to be specific about routines
which you could usefully apply. I hope others can provide this
information.

Best wishes,
Ted.


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E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 167 1972
Date: 02-Dec-03                                       Time: 14:25:52
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