# 1 Introduction

In MSEtool, assessment models are of class Assess. This appendix provides a brief description and references for the Assess objects. Further details regarding parameterization, e.g., fixing parameters, and tuning, e.g., adjusting start parameters, are provided in the function documentation.

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# 2 Surplus-production (SP) model

## 2.1 Dynamics equations

The surplus production model uses the Fletcher (1978) formulation. The biomass $$B_t$$ in year $$t$$ is $B_t = B_{t-1} + P_{t-1} - C_{t-1},$ where $$C_t$$ is the observed catch and $$P_t$$ is the surplus production given by: $P_t = \gamma MSY \left(\dfrac{B_t}{K}-\left[\dfrac{B_t}{K}\right]^n\right),$ where $$K$$ is the carrying capacity, $$MSY$$ is the estimated maximum sustainable yield, and $$n$$ is the parameter that controls shape of the production curve, and $$\gamma$$ is $\gamma = \dfrac{1}{n-1}n^{n/(n-1)}.$

By conditioning the model on observed catch, the predicted index $$\hat{I}_t$$ is $\hat{I}_t = \hat{q} \hat{B}_t$ and the harvest rate is $\hat{F}_t = \dfrac{C_t}{\hat{B}_t}.$ The dynamics equations above use an annual time step. Optionally, smaller time steps are used in the model to approximate continuous production and fishing. Given the biomass in the start of the year and assuming a constant fishing mortality over the time steps within a year, the fishing mortality that produces the observed annual catch is solved iteratively.

The likelihood of the observed index $$I_t$$, assuming a lognormal distribution, is $\log(I_t) \sim N(\log[\hat{I}_t], \sigma^2).$

## 2.2 Derived parameters

From estimates of leading parameters $$F_{MSY}$$ and $$MSY$$, the biomass $$B_{MSY}$$ at $$MSY$$ is $B_{MSY} = \dfrac{MSY}{F_{MSY}},$ the carrying capacity $$K$$ is $K = n^{1/(n-1)} B_{MSY} ,$ and the intrinsic rate of population increase $$r$$ is $r = n F_{MSY}.$ The production parameter $$n$$ is typically fixed and the model has a symmetric productive curve ($$B_{MSY}/K = 0.5$$) when $$n = 2$$.

## 2.3 State-space version (SP_SS)

In the state-state version, annual biomass deviates are estimated as random effects. Similar to Meyer and Millar (1999), the biomass $$B_t$$ in year $$t$$ is $B_t = (B_{t-1} + P_{t-1} - C_{t-1})\exp(\delta_t - 0.5 \tau^2),$ where $$\delta_t \sim N(0, \tau^2)$$ are biomass deviations in lognormal space and $$\tau$$ is the standard deviation of the biomass deviations.

The log-likelihood of the estimated deviations $$\hat{\delta}_t$$ is $\hat{\delta}_t \sim N(0, \tau^2).$

# 3 References

Fletcher, R.I. 1978. On the restructuring of the Pella-Tomlinson system. Fishery Bulletin 76:515-521.

Meyer, R., and Millar, R.B. 1999. BUGS in Bayesian stock assessments. Canadian Journal of Fisheries and Aquatic Science 56:1078-1086.