Shareef,
You forwarded your reply, to my note, to FishFolk but not to the
FishSci list on which I posted it. I'll copy this response to both
lists, at some expense of bandwidth (and with apologies to those
FishFolk who find this excessively technical).
You wrote:
> > I have used Pauly's equation a number of times on fish stocks. I fully agree with
> > you on your criticism for using this equation on non teleosts. This equation
> > expresses, M (natural mortality) = f(growth rate, average maximum length (or
> > maximum weight), temperature), where M is estimated to be an average value.
I am sorry but Pauly's equation does not involve a function of "growth
rate". It uses the "k" parameter from the von Bertalanffy equation but
that is a measure of the rate of approach to the asymptotic size, not
the growth rate. (The former would be in units of "% of asymptote per
unit time", rather than "centimetres per unit time".)
The difference is important for two reasons:
(1) Fish that have evolved to approach their maximum size quickly are
probably also evolved for short lives and hence high mortality rates. A
supposed linkage between growth rate and mortality rate would imply some
much more complex underlying relationship, likely to do with rates of
energy flow.
(2) Dan Pauly did not have any true values of growth rates or maximal
sizes. He only had estimates of von Bertalanffy parameter values, along
with those of water temperature and natural mortality rate, extracted
from other people's publications. When you fit the von Bert. function to
a particular data set, it generates hyperbolic confidence intervals
since the estimate of "k" is inversely related to the estimate of the
asymptotic size. Often both are wildly wrong but their ratio is
precisely estimated. By including both in his equation, Dan did not need
exact values of "k" or L[inf]  if one was too large, the other would
be too small and a straight line could still pass through the plotted
point. Had he used growth rate instead, his fit (and all of our
subsequent uses of the equation to estimate M) would have risked being
seriously erroneous.
> > The
> > above functional relation indicates that natural mortality is largely governed by
> > size and temperature.
On that too, I must disagree most strongly. Dan's equations (both
versions) provide a descriptive model by which M can be estimated. I see
no grounds for supposing that it is a mechanistic model and hence none
for supposing that mortality rate (the dependent variable in the
regression) is determined by the three variables that Dan chose to use
as independents. His choice of them was heavily influenced by the
availability of those particular pieces of information for many stocks
and, had there been such informaton for others, I dare say he would have
included it too. Would that make the other variable (water depth or
fecundity perhaps?) into a factor that governs mortality rate?
You could just as well suppose that mortality rate governs growth
patterns: Species with long lives can afford to take a long time to
reach large sizes before devoting most of their energy intake to reproduction.
> > The equation works satisfactorily with teleosts, which
> > grow continuously throughout its life span. When we consider non teleosts, where one
> > has to use legnthbased models mostly, the growth is not continuous and sometimes
> > may stop at some stage. So, if one takes fish ideas (like this one)
> > directly to invertebrate modeling, he or she is going to face problems. This is
> > the main reason for dragging Pauly's equation to the scene.
I agree that noncontinuous growth (or continuous growth that does not
follow a von Bert. curve) poses problems for Dan's equation. There are,
however, a lot of invertebrates that do approximate to von Bert. I still
wouldn't use Dan's equations with those. As I indicated before, I
wouldn't even use it with sharks.
> > So, let me repeat my
> > question once again,
> >
> > Can we incorporate submodels describing natural mortality and maturity processes
> > directly in terms of length in a lengthbased model?
I specifically avoided that question in your original post and tackled
only your subsidiary inquiry. If I cannot so easily avoid it:
If your lengthbased model is otherwise valid (a big "if"), of course
you can incorporate natural mortality and maturity into the model. What
you might not be able to do is to incorporate maturity by a "knifeedge"
function but, for many purposes, you cannot do that in an agebased
model either.
If natural mortality is treated as a constant M, you would need to make
some allowance for most fish taking more time to increase by one size
increment when they are large than when they are small. Thus, the
natural mortality per length step would increase with fish size, even
though the rate per time step was steady.
If you are concerned to model changes in the natural mortality rate
through life: Declining mortality rates in young fish could be modeled
as easily by length as by age since fish growth is quasi linear at those
ages/sizes. If you have reason to suppose that a species suffers
senescence, you need some model of the change in M with size/age. That
is likely to be much more of a good guess than is any growth curve for
the same species. Thus, a model of M increase with size can be converted
into one of M increase in age without materially lowering the (already
low) precision of the model.
In short, I think you are concerned over one of the least of your
problems, instead of focusing on the major challenges.
Trevor Kenchington

Trevor J. Kenchington PhD [log in to unmask]
Gadus Associates, Office(902) 8899250
R.R.#1, Musquodoboit Harbour, Fax (902) 8899251
Nova Scotia B0J 2L0, CANADA Home (902) 8893555
Science Serving the Fisheries
http://home.istar.ca/~gadus
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