27  Population Dynamics

Note: no simulation subscript in any equations

27.1 Number at Age

27.1.1 Initial

  • annual
  • seasonal - initialized to account for seasonal recruitment

Notes:

  • age of recruitment is MinAge(Stock)
  • MinAge(Stock) == 0 means that recruitment occurs in same time step as SProduction
  • otherwise lagged by appropriate number of time steps (year, quarters, months)

\[ N_{t+1} = N_t ... \]

28 Spatial Distribution of Fishing Effort

Note: If users provide values for Distribution for any specific sims, fleets, and years, these values will be used instead of the calculations below

28.1 Spatial Utility

Note: subscripts follow the structure of the arrays in the model

Spatial utility is currently calculated as proportional to fleet- and area-specific expected catch rates (currently assuming a unit of catch in weight is equally valuable for all stocks), accounting for within-time-step depletion.

Exploitable biomass per unit effort \(B\) for stock \(s\) and fleet \(f\) in year \(y\) and area \(r\) calculated as the fleet-specific vulnerable biomass multiplied by fleet-specific catchability:

\[ B_{s,y,f,r}=q_{s,y,f}\sum_{a}N_{s,y,a,r}\, W_{s,y,a,f}\, S_{s,y,a,f,r}\, R_{s,y,a,f,r} \]

where:

  • \(q_{s,y,f}\) : catchability by or stock \(s\) an fleet \(f\) in year \(y\)

  • \(N_{s,y,a,r}\): number-at-age \(a\) for stock \(s\) in year \(y\) and area \(a\)

  • \(W_{s,y,a,f}\): weight-at-age \(a\) for stock \(s\) and fleet \(f\) in year \(y\)

  • \(S_{s,y,a,f}\): selectivity-at-age \(a\) for stock \(s\) and fleet \(f\) in year \(y\)

  • \(R_{s,y,a,f}\): retention-at-age \(a\) for stock \(s\) and fleet \(f\) in year \(y\)

Note: the above equation could/should include (age-specific) value per unit catch biomass in the summation term to account for stock-specific targeting/preference. Current model assumes all stocks have equal value per unit catch

Area-specific fishing utility is assumed to be proportional to exploitable biomass, with a fleet saturation term that scales with biomass density relative to effective habitat capacity, ensuring that small areas saturate more rapidly under fishing pressure. The stock-specific utility is calculated as:

\[ \hat{U}_{s,y,f,r}=\frac {B_{s,y,f,r}} {1+\phi_{s,y,f}\Gamma_{s,y,f,r}} \]

where \(\phi_{s,y,f}\) represents the relative fishing pressure \(q_{s,y,f}E_{y,f}\) and \(\Gamma_{s,y,f,r}\) represents the relative density of exploitable biomass:

\[ \Gamma_{s,y,f,r} = \begin{cases} \frac {B_{s,y,f,r}} {A_r B^\text{ref}_{s,y,f}} & B^\text{ref}_{s,y,f}>0 \\ 0 & \text{otherwise} \end{cases} \]

where \(A_r\) is the relative size of area \(r\) and \(B^\text{ref}_{s,y,f}\) is calculated as the median exploitable biomass across all areas and used as a normalization that is insensitive to extreme local biomass values:

\[ B^\text{ref}_{s,y,f} = \text{median}_r\left( B_{s,y,f,r}\right) \]

Relative spatial utility is then calculated by summing \(\hat{U}_{y,f,r}\) over stocks and normalizing:

\[ U_{y,f,r} = \frac{ \sum_s{\hat{U}_{s,y,f,r}}} {\sum_r\sum_s{\hat{U}_{s,y,f,r}}} \]

Note that the model currently assumes all stocks have equal value per unit catch weight, and all areas have relative cost per fleet (and stock). This model can be extended later to include price (value) of catch-at-age/size, cost-per-unit-effort for each fleet, fixed spatial costs, etc

Note: currently there is no stochasticityin utility - this could be added

28.2 Distribution of Fishing Effort

Note: currently there is no constraint on the maximum effort per area - this could be added as a parameter in the fleet object.

The spatial distribution of fishing effort is calculated as:

\[ E_{y,f,r}=E'_{y,f}P_{y,f,r} \]

where \(E'_{y,f}\) is total effort for fleet \(f\) in year \(y\) and \(P_{y,f,r}\) is the probability of effort calculated using a gravity model:

\[ P_{y,f,r}=\frac{U^\theta_{y,f,r}} {\sum_{r'}{U^\theta_{y,f,r'}}} \text{with } U_{r'} \geq 0 \]

where \(P_{y,f,r}\) is the probability of fleet \(f\) fishing in area \(r\) in year \(y\) and \(\theta\) is a effort concentration or spread parameter:

  • \(\theta = 0\) : effort uniformly distibuted

  • \(\theta = 1\) : effort proportional to utility

  • \(\theta < 1\): effort more evenly distributed than utility

  • \(\theta > 1\): effort more concentrated than utility

Note: additional models could include logit random utility model