Mackerel life history parameters are used to develop an example of modelling density dependence for mass, maturity and natural mortality-at-age. The parameters are first used to construct An FLBRP object representing the equilibrium, and are then coerced into an FLStock to model the time series dynamics.

Mass-at-age

Mass-at-age (\(W_a\)) is modelled as

\(W_a = {\alpha}_a B^{{\beta}_a}\)

Where \(\alpha\) and \(\beta\) are estimated from a regression of mass-at-age (\(W_a\)) on total biomass (\(B_a\)), based on empirical data. \(W_a\) is constrained so sizes do not become unfeasible large or small.

Maturity-at-weight

Maturity is modelled as a logistic function of mass-at-age i.e.

\(O(W_a) = 1/(1+e^{k(W_a-W_{50})})\)

Natural mortality

Is assumed to vary at unit length (\(L_1\)) e.g.

\(ln(M_1) =\alpha + \beta ln(L_{\infty}) + \gamma ln(k)\)

where \(\alpha=0.65\), \(\beta=0.91\), \(\gamma=0.87\)

Length-at-age (\(L_a\)) can be obtained from the length-weight relationship

\(L_a=(W/a)^{1/b}\)

giving

\(ln(M_1) =(W/a)^{1/b}(\alpha + \beta ln(L_{\infty}) + \gamma ln(k))\)

For DD there is a relationship between \(M_a\) and total biomass (\(B\))

Mackerel:

\(M = 0.0232B + 0.0524\)

Blue whiting:

\(M = 0.0221B + 0.0719\)

Combining these provides values of \(M_a\)

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Simulations

Figure 1 Density dependence multiplier, red is \(B_{MSY}\)

Figure 2 Simulation of density dependence in mass-at-age

Figure 3 Simulation of density dependence in maturity-at-age

Figure 4 Pope: simulation of density dependence in M-at-age

Figure 5 Lorenzen: Simulation of density dependence in M-at-age

Figure 6 Mass-at-age: Comparison of equilibrium curves with (black) and without (red) density dependence.

Figure 7 Mass and maturity-at-age: Comparison of equilibrium curves with (black) and without (red) density dependence.

Figure 8 Mass, maturity and M-at-age: Comparison of equilibrium curves with (black) and without (red) density dependence.

Figure 9 Mass, maturity and M-at-age: Comparison of projections with and without density dependence.

Funding

References

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