The calculation of production, reproduction and total growth in the autonomous Vansickle equation

Summary

• 1. The VanSickle equation is derived from the Von Foerster equation by a change of variable from age to weight with growth not depending on time explicitly.

• 2. The author's, previous treatment of Pechen's first graphical method for calculating production is corrected.

• 3. Pechen's estimate, here called total growth, is reduced to one integration.

• 4. Production (Thienemann's definition) amounts the calculation of one Stieltjes integral.

• 5. Production, thus defined, equals reproduction in weight plus total growth (Pechen's estimate).

• 6. The fact that reproduction is continuous allows an easy estimate of the birth-rate from the age distribution of the number of organisms, by using the Von Foerster boundary condition.

• 7. The Von Foerster functional boundary condition is consistent with a particular reproductive mechanism called “differentiable reproduction”, in which no discontinuities in the organism's weight because of births are allowed.

• 8. Numerical calculations on Pechen'sCyclops data are explicitly represented. Production comes at 18.4 instead of 16.6. An averageCyclops individual may spend 13% of its production on reproduction.

• 9. The calculation of production with the aid of the Stieltjes integral shows that the tail of the age or weight distribution comprising the oldest individuals might be very important.

• 10. Finally there is a discussion on open problems; mathematically speaking there is still a tremendous lot to be done.