Windows of Opportunity for Sustainable Fisheries Management: The Case of Eastern Baltic Cod

Abstract

We study under which conditions a ‘window of opportunity’ for a change from an overfishing situation, with high fishing effort, but low stocks and catches, towards sustainable fishery management arises. Studying the Eastern Baltic cod fishery we show that at very low stock sizes (as they prevailed in the early 2000s) all interest groups involved in the fishery unanimously prefer maximum-sustainable-yield management (as prescribed by the management plan in place since 2007) over the previous overfishing situation. With increasing stock sizes, the present value of fishermen surplus would be higher when switching back to overfishing again, while other interest groups maintain their preference for sustainable fishery management.

Appendix

1.1 Conversion of an Age-Structured Model into a Biomass Model

To set up the age-structured model, we use a similar approach and notation as Tahvonen (2009) and Tahvonen et al. (2013). We use $x_{st}$ to denote the number of fish in age group $s=1,\ldots ,\, S$ at the beginning of period $t=0,\, 1,\ldots$, where S is the oldest age group considered in the model. Age-specific survival rates $\alpha _{s} >0$, age-specific proportions of mature individuals $\gamma _{s} >0$, and mean weights of fish (in kilograms) $w_{s}$, in age groups $s=1,\ldots ,\, S$, all are assumed to be constant as in the standard biological stock assessments, such as ICES (2012) for the Eastern Baltic cod.

Harvesting takes place at the beginning of each period but after recruitment. The fisheries literature typically uses the instantaneous fishing mortality $f_{t}$. Assuming that the instantaneous fishing mortality $f_{t }$ is constant throughout the fishing season, we can use the Baranov catch equation that gives the fraction of the stock harvested during the entire season, i.e. the exploitation rate, as $1-\mathrm{exp}\left( -f_{t} \right)$. Note that the instantaneous fishing mortality, i.e. the instantaneous exploitation rate, can be above one. Only in the limit $f_{t} \rightarrow \infty$ the number of fish harvested over the entire fishing season would equal the stock size. The age-structured population model with harvesting activity is described by the following equations

$$\begin{aligned} x_{0t}&=\sum \limits _{s=1}^{S} \gamma _{s} \, w_{s} \, x_{st} \nonumber \\ x_{1,\, t+1}&=\varphi (x_{0t} ) \nonumber \\ x_{s+1,\, t+1}&= \alpha _{s} \, \left( x_{st} -(1-\exp (-q_{s}^{} f_{t} ))\, x_{st} \right) \, \text { for }s=1,\ldots ,\, S-2 \\ x_{S,\, t+1}&= \alpha _{S-1} \, \left( x_{S-1,\, t} -(1-\exp (-q_{S-1}^{} f_{t}^{} ))\, x_{S-1,\, t} \right) \nonumber \\&\quad +\alpha _{S} \, \left( x_{St} -(1-\exp (-q_{S}^{} f_{t} ))\, x_{St} \right) .\nonumber \end{aligned}$$

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The spawning stock biomass $x_{0t}$ is given by the sum of biomasses (age-specific weight $w_{s}$ times number of fish in that age group, $x_{st}$) of the fraction of fish that is mature in the respective age classes, $\gamma _{s}$. The number of recruits, $x_{1t}$, is given by the stock-recruitment function $\varphi (x_{0t} )$. Of all fish of age group s that remain in the sea after fishing, a fraction $\alpha _{s}$ survives natural mortality (second last of the above equations). The equation describing the dynamics for the last age group differs from the equations for the younger age groups, as $x_{St}$captures all individuals that are of age S and older in year t.

For the Eastern Baltic cod stock, we consider eight age classes ($S=8$), as in the standard stock assessment by the International Council for the Exploration of the Sea (ICES 2012). We parameterize the age-structured fish population using data from the ICES (2012) assessment report. For age-specific weights in stock $w_{s}$ we use the values for 2011; also age-specific maturities $\gamma _{s}$ are directly taken from the assessment report. The survival rates $\alpha _{s}$are computed from natural mortalities. To account for age-dependent vulnerability to fishing gear, we multiply the fishing mortality rate with a catchability factor for each age class. For the Eastern Baltic Cod stock, these catchability factors are computed from the age-specific fishing mortalities averaged over the 5 years from 2002 to 2006. The age-specific parameter values are summarized in Table 1.

Table 1 Parameter values for age-structured fish population model from ICES (2012)

We assume a stock-recruitment function of the Ricker (1954) type,

$$\begin{aligned} \phi (x_{0} )=\beta _{1} \, x_{0} \, e^{-x_{0} /\beta _{2} }. \end{aligned}$$

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This stock-recruitment function has a peak at $x_{0} =\beta _{2}$, and is decreasing for spawning stocks larger than $\beta _{2}$. Such a type of stock-recruitment relationship is an appropriate description of recruitment biology of Baltic cod, capturing in particular the cannibalism of older cod on juveniles. For the parameters of the stock-recruitment function, we use the estimates $\beta _{1} =1.70$ and $\beta _{2} =1/0.00182=549,000$ tons from Voss et al. (2014).

In order to transform the eight-dimensional population model into a biomass model, we compute the steady-state stock biomass and harvest biomass for different fishing mortality rates. We then use the equilibrium harvest and biomass from the age-structured population model to fit the surplus production function (Blenckner et al. 2015)

$$\begin{aligned} g(B_{t}^{} )=\frac{r}{\alpha } B_{t}^{} \left( 1-\left( \frac{B_{t}^{} }{K} \right) ^{\alpha } \right) \end{aligned}$$

We obtain estimates $r=0.736$, $K=1158$ thousand tons, and $\alpha =1.441$. The graph of the resulting surplus-production function is shown in Fig. 1.