# Model And Calibration • 3 species: Cod, Sprat, Herring

3 species : Cod, Sprat, Herring

The Age-structured Model

this model  takes into account the predator-prey relationship between cod and the two other species and is discrete-time, discrete-aged structured.

$\begin{array}{rl}{x}_{i0t}& =\sum _{s=1}^{S}{\gamma }_{is}\phantom{\rule{thinmathspace}{0ex}}{w}_{is}\phantom{\rule{thinmathspace}{0ex}}{x}_{ist}\end{array}$$\begin{array}{rl}{x}_{i1,t+1}& ={\phi }_{i}\left({x}_{i0t}\right)\end{array}$$\begin{array}{rl}{x}_{i,S,t+1}& ={\alpha }_{i,S-1}\phantom{\rule{thinmathspace}{0ex}}\left({x}_{i,S-1,t}-{h}_{i,S-1,t}\right)+{\alpha }_{iS}\phantom{\rule{thinmathspace}{0ex}}\left({x}_{iSt}-{h}_{iSt}\right).\end{array}$

where xist denotes  the number of fish of species i∈{C,S,H}, where C stands for cod, S for sprat, and H for herring, in age group s=1,…,S and at the beginning of period t=0,1,…t.
αis>0 denotes age-specific survival rates,
γis>0 denotes age-specific proportions of mature individuals,
ωis to denote the mean weights (in kilograms). For cod, all of these parameters are assumed to be constant . For sprat and herring, we assume that proportions of mature individuals and weights are constant, but the survival rates depend on cod spawning stock biomass.
The recruitment function for species i is denoted by φi(⋅) and the spawning biomass by xi0t, hist denotes the number of fish harvested from cohort s of species i in period t.

The modeling of profits of the cod fishery, is based on the specification from Quaas et al. (2012) with age-specific prices and a cost function of the Spence type .
Thus, profits of the cod fishery in year t are: