Instead of developing a detailed explicit food web through the whole pelagic ecosystem, which is a complex and very long task, the tuna forage is considered as a single population. Since all the organisms of a same cohort have the same age, it is immaterial whether the cohorts are of the same species or not.

The classical fish population approach is used to describe the dynamic of this forage population, with:
a continuous recruitment S

a mortality coeffcient l

The development of a new primary biomass during a period of time (primary production) allows the development of a new cohort of many different organisms, a fraction of which will be recruited after a time Tr in the forage population. The level of fraction transferred depends of an ecological transfer coefficient. Since the cohorts are identical in their growth and mortality, it is not necessary to consider effects due to the existence of more than one cohort. The total biomass integral by time of a single cohort is equal to the total biomass of the population of successive and identical cohorts. In an equilibrium situation, the biomass F~S/l and the production F’=S. 

Transfer with time of primary production towards forage according to the model (S is assumed constant). The thin curve describes the evolution in time of a single source of primary production. The thick curve gives the total forage population.

Assuming that the recruitment S is constant and continuous allows defining a "mean age" to the forage population. If  Tr is at time 0 of the forage population, the sum of the products t.St is: r is at time 0 of the forage population, the sum of the products t.St is:

Since the forage population is S/l, the mean time spent in the forage population is S/l² divided by S/l, or 1/l. Adding T_r to 1/l gives the "mean age" of the forage population since the origin of the cohort, i.e., the time when the new primary production appears. Specifically, it is a biomass-weighted mean age because the forage population is expressed in unit of biomass. The mean age can be assimilated to the interval between one generation and the next, such as a mean generation time or turnover time (the mean time taken for the biomass of the population to be replaced by fresh production). It is also possible to define the maximum life span of organisms within the forage population as the time necessary to see the population reduced by a determined level (e.g., for 99%, t = - 1/l . Ln(0.01) + T_r). Therefore, l and T_r which characterize F can be estimated using biological characteristics of key-species representative of the forage population.

Application to tuna forage

Tunas are known to be opportunistic predators feeding upon three major groups of prey: fish, squids and crustaceans. Sizes of prey items range from a few millimeters (small euphausids and amphipods) to several centimeters (squids, small fish, shrimps). Biological characteristics of the main tuna prey species or group of species can be found in the literature. A mean age of 150 d was chosen with a value of 60 d for Tr. These parameter values lead to a maximum life span of 474 d. Although these values may need to be more accurately refined, they appear to be a reasonable characterization of tuna forage according to biological studies of typical prey species.

To follow the energy transfer from primary production to forage, the nitrogen unit of primary production is also used for the forage population. Therefore, the forage biomass is the accumulated quantity of nitrogen in the population. At this stage, there is no growth function for the forage population, but an appropriate coefficient is used to convert this quantity of nitrogen to wet weight. In future development however, a growth function based on the d15N content of the prey items of tuna could be used.

To follow the energy transfer from primary production to forage, the nitrogen unit of primary production is also used for the forage population. Therefore, the forage biomass is the accumulated quantity of nitrogen in the population. At this stage, there is no growth function for the forage population, but an appropriate coefficient is used to convert this quantity of nitrogen to wet weight. In future development however, a growth function based on the d15N content of the prey items of tuna could be used.

Spatial dynamic of forage population

For the movement of small forage organisms and tuna larvae and juveniles, the advective components in the two horizontal dimensions are oceanic currents, with r, the diffusion coefficient (diffusion of water and random movement of organisms), and u, v the zonal and meridional components of the current in the euphotic layer. F is the forage, S the new organisms recruited in the forage population and l the mortality coefficient.

References

Isaacs, J.D. (1977). The life of the open sea. Nature, 267, 778-780
Sheldon, R.W., Prakash, A., & Sutcliffe Jr, W.H. (1972). The size distribution of particles in the Ocean. Limnology Oceanography, 17, 327-340