Hi Colleague,

I would think that the size-based models are the same as age-based ones.
Here I would like to share with you a very simple example of this (I have
given slightly more involved demonstrations here and there and found this
simple one works up to now). I understand many of you would know it, in
which case you just hit the delete key. I also acknowledge first that this
idea may not be new.

Now, let's have the usual simple tille population dynamics equation (number
dynamics)
dN(t)/dt= -Z(t) N(t)         (1)
and size dynamics equation
dL(t)/dt=f(L(t))              (2)
where N(t) is the number at age t, Z(t) is the total mortality (natural
mortality plus fishing mortality), L(t) is a size measure of an animal, and
f(L(t)) is some function of L(t).

The solution of (1) as an intial value problem is
N(t)=N(t0)*exp(-integrate Z(s) with respect to s from t0 to t)        (3)
where t0<=t is some meaningful age. You can call (3) the solved number
dynamics equation.

Now, to show both types of models are equivalent, divide (1) by (2), which
gives
dN(t)/dL(t)= -Z(t)/f(L(t)) N(t),        or
dN(t)/N(t)= -Z(t)/f(L(t)) dL(t)        (4)
integration of which with respect to L(t) gives a size-based model, i.e.,
N(L(t))=N(L(t0)) exp(- integrate Z(t(s))/f(s) with repect to s from L(t0) to
L(t))    (5)
where t in the integrand is written explicitly as a function of L(t).

Eqn 4 or 5 is a size-based model; Eqn 1 or 3 is an age-based model;  Eqn 2
is just a growth curve.

Now, it is clear that equation 4 or 5 is implied by, or is equivalent to,
equations 1 and 2. In other words,
Size-based models=age-based ones -a growth in size curve,   or
Age-based models =size-based ones +a growth in size curve.

One more thing you might appreciate. Notice the growth rate in size in
equation 4 or 5. This fact implies, of course, that one cannot simply set up
a size-based model, without considering growth in age. [I can now hear your
crying, because you cannot age your animal, because it does not have a
backbone. I am sorry] This is because growth rate in AGE has to be in the
model as in equation 4 or 5. It is not very wise to just stick the Z(t)
there and forget about its beautiful denominator, because in so doing one
assumes that the rate of growth in size is unity (i.e., 1, or one). After
all, if there is one on top, then there must be one down, which is my
version of biological relativity!

Jokes aside. If you are prepared to accept that assumption, bussiness is to
be as usual. After all, it is just one single term, although it would be
very nasty if f(L(t)) is close to, or equal to, zero.

I hope this simple explanation makes sense.


Best wishes

Yongshun Xiao
SARDI Aquatic Sciences Centre
2 Hamra Avenue, West Beach
Adelaide, SA, Australia 5024
Email [log in to unmask]
Phone + 61 8 8200 2434
Fax   + 61 8 8200 2481

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