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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