Hi Colleague,
I would think that the sizebased models are the same as agebased 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 sizebased 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 sizebased model; Eqn 1 or 3 is an agebased 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, Sizebased models=agebased ones a growth in size curve, or Agebased models =sizebased 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 sizebased 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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