Fishing, reproductive volume and regulation: population dynamics and exploitation of the eastern Baltic cod

Abstract

The relative importance of exploitation rate and environmental variability in generating fluctuations of harvested populations is a key issue in academic ecology as well as population management. We studied how the eastern Baltic cod (Gadus morhua) is affected by fishing and environmental variation by using a newly developed single species state-space model. Survey data and auxiliary environmental data were used to estimate the model parameters. The model was then used to predict future development of the eastern Baltic cod under different fishing mortalities and abiotic conditions. Abiotic condition was represented by an index: reproductive volume which is the volume of water suitable (in terms of salinity and oxygen content) for the successful development of the early life stages of Baltic cod. The model included direct density dependence, fishing, and a lagged effect of reproductive volume. Our analysis showed that fishing rate is approximately three times more important than reproductive volume in explaining the population dynamics. Furthermore, our model suggests either under- or over-compensatory dynamics depending on the reproductive volume and long term catch levels. It follows that fishing can either reduce or increase temporal oscillations of the cod stock depending on whether the dynamics is over- or undercompensatory, respectively. The sustainable level of fishing rate is however dependent on reproductive volume. Our model predicts a dual role of fishing rate, stabilizing when reproductive volume is high and destabilizing when it is low. Exploitation rate may therefore increase or decrease the risk of the population of cod dropping below a given biomass reference point depending on the environmental conditions, which has practical implications for fisheries management.

Appendix

Reproductive volume, dissolved oxygen or salinity as environmental covariate

We examined reproductive volume (RV _t ), dissolved oxygen and salinity (from the Gotland basin 80–100 m in depth; SMHI 2010) as possible environmental covariates to explain changes in cod biomass. However, according to Table 1 salinity and dissolved oxygen did not have any predictive power in our dataset and we restricted the analysis to RV _t .

A relation between index and total stock biomass

The SSB threshold level for the Eastern Baltic cod is estimated as the lowest level of SSB under which recruitment is considered impaired (ICES 2010). The latest available threshold value (SSB _threshold) was in 2008 estimated by ICES at around 230,000 tonnes (ICES 2008). To be able to compare our result with recommendations from the ICES we therefore transformed their estimates of spawning stock biomass to an index according to:

$$I_{t} = qSSB_{t}^{\alpha }$$

(5)

In logarithm form this becomes:

$${ \log }_{\text{e}} \left( {I_{t} } \right) = { \log }_{\text{e}} \left( q \right) \, + \alpha { \log }_{\text{e}} \left( {SSB_{t} } \right)$$

(6)

SSB _t is total spawning stock biomass, I _t is the cod index, q and α are constants. To estimate q and α we regressed log_e(I _t ) on log_e(SSB _t ) using data from the time period 1991–2010. As the only available data on SSB _t in the Baltic Sea originated from a VPA we used this source despite all potential problems pointed out earlier in this work. The point estimates are: q = 0.18 and α = 0.60.

The minimum sustainable spawning biomass (SSB _threshold) for the Eastern Baltic cod is, according to ICES, 230,000 tonnes. Equation 5 with the estimated values of q and α gives the corresponding value: I _threshold = 310 (number of cod caught per hour).

Model dynamics

To better understand the dynamic behaviour of Eq. 4 when simulated for different (fixed) values of catch (C _t  = C) and reproductive volume (RV _t  = R), we express the model in general terms:

$$J_{t + 1} = f(J_{t} ,\varepsilon_{t} )$$

(7)

and linearize around the deterministic equilibrium J ^* evaluated at the expected values of the noise, E(ε _t ) = ε ^* = 0:

$$J_{t + 1} - J^{*} = \frac{{\partial f(J^{*} ,\varepsilon^{*} )}}{\partial J}\left( {J_{t} - J^{*} } \right) + \frac{{\partial f(J^{*} ,\varepsilon^{*} )}}{\partial \varepsilon }\varepsilon_{t}$$

(8)

Denoting the partial derivatives:

$$a = \frac{{\partial f(J^{*} ,\varepsilon^{*} )}}{\partial J}$$

(9)

$$b = \frac{{\partial f(J^{*} ,\varepsilon^{*} )}}{\partial \varepsilon }$$

(10)

Equation 8 can be written

$$x_{t + 1} = ax_{t} + b\varepsilon_{t}$$

(11)

The parameter a determines the endogenous dynamics, which is under-compensatory if a > 0 and over-compensatory if a < 0. By substituting γ = α ₁ + α ₃ C + α ₄ R we find that f = J _t exp(γ + α ₂ J _t  + ε _t ), J ^* = -γ/α ₂, a = 1-γ and b = -γ/α ₂. Because α ₃ < 0 and α ₄ > 0 (see Table 3), increased reproductive volume makes the dynamics more over-compensatory, whereas this effect is counteracted by fishing. Equation 11 is a first-order autoregressive process (e.g., Box and Jenkins 1976) its variance equals to:

$$V(x_{t} ) = \frac{{b^{2} V(\varepsilon_{t} )}}{{1 - a^{2} }} = \frac{{\gamma^{2} V(\varepsilon_{t} )}}{{\alpha_{2}^{2} (1 - (1 - \gamma )^{2} )}}$$

(12)

For small catches, such that:

$$C < C_{\text{stab}} = \, \left( { 2 { } - \alpha_{ 1} - \alpha_{ 4} R} \right)/\alpha_{ 3}$$

(13)

we have an unstable system and therefore large oscillations.

It should however be noticed that the stability limit above is for the deterministic part of the model. A nonlinear model with a noise disturbance can, if the perturbations are large enough, show a stable oscillation in the stable region but near the border of stability (se e.g., Kaitala et al. 1996). Our nonlinear stochastic model is therefore still oscillating for a slightly larger value of C than is postulated by Eq. 13. This can explain why the border of the area with the highest probability (P = 0.87) in Fig. 2b is approximately 10,000 tonnes to the right of the stability border defined by Eq. 13 (see Fig. 3b).

When:

$$C > C_{\text{zero}} = \, \left( { - \alpha_{ 1} - \alpha_{ 4} R} \right)/\alpha_{ 3}$$

(14)

the deterministic equilibrium value is zero (Fig. 4b).

The coefficient of variation is defined by:

$${\text{CV }} = \sqrt V /J^{*}$$

(15)

The value of C that minimises CV is defined by:

$$C_{\text{CVmin}} = \, \left( { 1 { } - \alpha_{ 1} - \alpha_{ 4} R} \right)/\alpha_{ 3}$$

(16)

C _CVmin also defines the border between under- or over-compensatory endogenous dynamics of the system.

To summarise the result:

0 < C < C _stab the system is unstable.

C _stab < C < C _CVmin the system is stable, over-compensatory and CV is decreased with increased fishing.

C _CVmin < C < C _zero the system is stable, under-compensatory and CV is increased with increased fishing.

C > C _zero the systems equilibrium value is negative which corresponds to an extinct population.

Model convergence

To test model convergence we first run simulations with two different start values for the model parameters: α ₁–α ₄:

Chain 1: α ₁ = 0.72, α ₂ = −0.0020, α ₃ = −0.41, α ₄ = 0.20

Chain 2: α ₁ = 2.00, α ₂ = −0.0050, α ₃ = −1.0, α ₄ = 0.10

The resulting chains can be seen in Fig. 5. The two chains look similar for all four parameters (no tendency of divergence).

Fig. 5

MCMC chains for model parameters. Two simulations with different start values were tested and the resulting 8 chains are: a α ₁ simulation 1, b α ₁ simulation 2, c α ₂ simulation 1, d α ₂ simulation 2, e α ₃ simulation 1, f α ₃ simulation 2, g α ₄ simulation 1, h α ₄ simulation 2

The second convergence test was a Gelman test (Brook and Gelman 1998). We used three different start values for the model parameters α ₁–α ₄, σ _obs and σ _proc. The potential scale reduction factor for the model parameters were:

α ₁ = 0.99995, α ₂ = 1.0001, α ₃ = 1, α ₄ = 1, σ _obs = 1.0006, σ _proc = 1.0003

A value near 1.0 for the potential scale factor indicates that the model has converged.

The third test was to check the sensibility for choice of prior distributions. We tested two additional distributions for σ _p and σ _m: half-t ₅ priors (original) uniform (0, 10) (test1) and inverse gamma (0.001, 0.001) (test2) (Table 2).

Our conclusion is that all three tests indicate that the model has converged.