Risk and Sustainability: Assessing Fishery Management Strategies

Abstract

We develop a theoretical framework to assess the sustainability of fishery management strategies, when the bioeconomic dynamics are marked by uncertainty and several conflicting objectives have to be accounted for. Stochastic viability ranks management strategies according to their probability to sustain economic and ecological outcomes over time. The approach is extended to build stochastic sustainable production possibility frontiers representing the trade-offs between sustainability objectives at any risk level, given the current state of the fishery. This framework is applied to a Chilean fishery faced with El Niño uncertainty. We study the viability of effort and quota strategies when catch and biomass levels have to be sustained. We show that (1) for these sustainability objectives, whatever the level of the outcomes to be sustained, quota-based management results in a better viability probability than effort-based management, and (2) the fishery’s historical quota levels were not sustainable given the stock levels in the early 2000s.

Appendix

1.1 Chilean Jack-Mackerel Case Study: Data, Parameters and Model

1.1.1 Historical Data for the Chilean Jack-Mackerel Fishery

Table 1 details the historical values of interest for the fishery.

Table 1 (a) DWFNs: Total annual catch of Distant Water Fishing Nations’ Fleets (fishing jack mackerel outside the Chilean EEZ). (b) The Chilean fleet’s TAC in column (3) is binding for catches within and beyond the Chilean EEZ. The first year to which TAC was applied in this fishery was 1999; the policy was resumed in 2001 (for more details see Gomez-Lobo and J. Pe na Torres, and P. Barria, (2011)). (c) To deduce the Chilean fleet’s (implicit) fishing effort multiplier ($\lambda$) in column (4), we replaced the annual catch $Y(N,\lambda )$ by its real historical values (column 1) in the Baranov equation (12) and simulated the stock dynamics: starting from the initial vector of abundances at age (for year 2002); we then applied the stock dynamics (equation 11) while considering the deterministic version of the Ricker recruitment function (equation 14), including the deterministic effect of El Niño events (in those years when it occurred, based on the definition in footnote 31). Sources: (1), (2), (5–7): IFOP (2013; 3): Subsecretaría de Pesca (Chilean Fisheries Regulator); (4): authors’ own calculations

1.1.2 Biological Model

We provide details of the model in Sect. 4.2.

The model is age-structured, with a Ricker stock-recruitment function. Abundance dynamics are given by

$$\begin{aligned} \left\{ \begin{array}{l} N_{a+1}(t+1) = \exp {(-(M_{a} + \lambda (t) F_{a}))} N_{a}(t) \;, \quad a=1,\ldots , A-2 \\ N_{A}(t+1) = \exp {(-(M_{A-1} + \lambda (t) F_{A-1}))} N_{A-1}(t) +\exp {(-(M_{A} + \lambda (t) F_{A}))} N_{A}(t) \end{array}\right. \nonumber \\ \end{aligned}$$

(11)

where $M_{a}$ is the natural mortality rate of individuals of age $a$, $F_{a}$ is the mortality rate of individuals of age $a$ due to harvesting between $t$ and $t+1$, supposed to remain constant during year $t$ (the vector $(F_{a})_{a=1,\ldots ,A}$ is termed the exploitation pattern).

Total annual catches $Y$, measured in million tons, are given by the Baranov catch equation (Quinn and Deriso 1999, pp. 255–256):

$$\begin{aligned} Y\bigl (N,\lambda \bigr )= \sum _{a=1}^{A} \varpi _{a}\, \frac{ \lambda F_{a}}{ \lambda F_{a} +M_{a}} \left( 1-\exp {(-(M_{a}+ \lambda F_{a}))}\right) N_{a} \, , \end{aligned}$$

(12)

where $(\varpi _{a})_{a=1, \ldots , A}$ are the weights at age.

The spawning stock biomass (SSB) is given by the expression

$$\begin{aligned} \textit{SSB}(N) = \sum _{a=1}^{A} \gamma _{a} \varpi _{a} N_{a} \; , \end{aligned}$$

(13)

where $(\gamma _{a})_{a=1, \ldots , A}$ are the proportions of mature individuals at age $a$ (some may be zero). Annual recruitment is a function of the SSB with a two-year delay, i.e., depending on the spawning stock biomass of two periods earlier:²⁸

$$\begin{aligned} N_1(t+1)=\alpha \textit{SSB}\big (N(t-1)\big )\exp \left( {\beta \textit{SSB}\big (N(t-1)\big )}+w(t)\right) \; , \end{aligned}$$

(14)

where $\{ w(t) \}$ is a random process reflecting the impact of climatic factors on the stock recruitment relationship (see below).

We use the parameter estimation proposed in Yepes (2004), which relies on official data from the Instituto de Fomento Pesquero (IFOP).²⁹ Parameters of the Ricker recruitment function at expression (14) were estimated using linear time-series analysis. The estimated parameters are $\alpha =e^{2.39}$ and $\beta =-2.2\cdot 10^{-7}$ (see Yepes (2004), p. 56). The values for parameters $M_a$ and $F_a$ are taken from IFOP’s official model for this fishery, so that $M_a$ is equal to 0.23 for all $a$ and $F_a$ is equal to the vector of averages values of $F_a$ during 2001–2002.³⁰

1.1.3 Stochastic Model

Following the statistical analysis in Yepes (2004), we simulate El Niño uncertain cycles using a sinusoidal function with random shocks.³¹ The random process $w(t)$ supposed to capture the effects of the El Niño phenomenon has a periodic part and an error term, $w(t) = -0.12 \times$ niño$(t)+ \epsilon (t)$, where

• the estimated error terms $\{ \epsilon (t) \}$ correspond to $\epsilon (t)=0.71\epsilon (t-1)-0.65\epsilon (t-2) + \mu (t),$ where $\{ \mu (t) \}$ is a sequence of i.i.d. random variables with Normal distribution $\mathcal{N}(0;0.18)$,

• niño$(t)= {\mathbf 1}_{ \{-1.2 \sin (18.19 + 2\pi (t -1951)/3.17) > 0.5\} }$ is a dummy (0 or 1) variable reflecting the presence of El Niño phenomenon.

1.1.4 Simulation Process

From a theoretical point of view, it is possible to determine the strategy that maximizes the viability probability by solving the dynamic programming equation characterizing the viability problem (De Lara et al. 2006). It is possible to obtain a closed-form solution for some problems (De Lara and Martinet 2009). Determining optimal strategies in dynamic optimization problems under uncertainty is not easy. Optimization in the stochastic viability framework is not exceptional. In particular, the curse of dimensionality can be a serious obstacle to the computation of optimal viability strategies.

From a practical point of view, it is possible to estimate the viability probability of any given strategy by means of Monte Carlo simulations. A random generator is used to produce scenarios following the distribution $\mathbb {P}$. For each scenario, a given management strategy is applied. If, for the corresponding trajectory, all the viability constraints in (4) are respected in each time period over the whole planning horizon, the scenario is viable for the applied management strategy. When the number of scenarios tested is large, the frequency of viable scenarios can be used as an approximation of the viability probability.