Optimal Harvesting Policies Threaten Biodiversity in Mixed Fisheries

Abstract

As marine ecosystems are under pressure worldwide, many scientists and stakeholders advocate the use of ecosystem-based approaches for fishery management. In particular, management policies are expected to account for the multispecies nature of fisheries. However, numerous fisheries management plans remain based on single-species concepts, such as maximum sustainable yield (MSY) and maximum economic yield (MEY), that respectively aim at maximizing catches or profits of single species or stocks. In this study, we assess the bioeconomic sustainability of multispecies MSY and MEY in a mixed fishery, characterized by technical interactions and therefore joint production. First, we analytically show how multispecies MSY and MEY can induce overharvesting and extinction of species with low productivity and low value. Second, we identify and discuss incentives on effort costs and landing prices, as well as technical regulations, that could promote biodiversity conservation and more globally sustainability. Finally, a numerical example based on the coastal fishery in French Guiana illustrates the analytical findings.

Appendix

1.1 A.1 Proof of Proposition 1

In this Appendix, we aim at solving the following optimization problem:

$$ \max_{e } \sum\limits_{i = 1}^{N} x_{i}^{+}(e) q_{i} e. $$

(31)

We consider that species are ranked according to their biotechnical productivities, as expressed in Eq. 6. Then, for a given effort e,¹

$$ \text{if} \exists j \in \{ 1 {\ldots} N \}, \text{so that} \frac{r_{j-1}}{q_{j-1}} \leq e < \frac{r_{j}}{q_{j}}, $$

(32)

as for every i < j, we have $x_{i}^{+}(e)= 0$, we deduce that

$$ \sum\limits_{i = 1}^{N} x_{i}^{+}(e) q_{i} e = \sum\limits_{i=j}^{N} x_{i}^{+}(e) q_{i} e $$

(33)

As a result, for a given effort e, total catches can be expressed as follows:

$$\begin{array}{@{}rcl@{}} \sum\limits_{i = 1}^{N} x_{i}^{+}(e) q_{i} e &=& \sum\limits_{i=j(e)}^{N} x_{i}^{+}(e) q_{i} e, \text{with} \\ j(e)&=& \min \left( i \in \{ 1 {\ldots} N \}, e < \frac{r_{i}}{q_{i}} \right). \end{array} $$

(34)

Therefore, optimization problem (31) can be written as follows:

$$ \max_{e } \sum\limits_{i = 1}^{N} x_{i}^{+}(e) q_{i} e = \max_{e} \sum\limits_{i=j(e)}^{N} x_{i}^{+}(e) q_{i} e, $$

(35)

where j(e) is defined in Eq. 34. Denote by e^∗ an optimal solution of this problem, and by j^∗ = j(e^∗) the corresponding first non-extinct species. By definition, optimal solutions verify the following condition:

$$ \frac{r_{j^{*}-1}}{q_{j^{*}-1}} \leq e^{*} < \frac{r_{j^{*}}}{q_{j^{*}}}. $$

(36)

As j^∗ is the surviving species with the lowest biotechnical productivity, species j^∗ to N are not extinct at MMSY, and their biomasses are thus larger than zero. The optimization problem (31) is then equivalent to:

$$ \max_{e} \left( \sum\limits_{i=j^{*}}^{N} \left( \frac{r_{i} - q_{i} e}{s_{i}} \right) q_{i} e \right), $$

(37)

The optimal effort can be computed by using first-order optimality conditions:

$$ \frac{\partial \left( {\sum}_{i=j^{*}}^{N} \left( \frac{r_{i} - q_{i} e}{s_{i}} \right) q_{i} e \right)}{\partial e} = 0. $$

(38)

We obtain

$$ e^{*} = \frac{1}{2} \frac{{\sum}_{i=j^{*}}^{N} r_{i} q_{i} s_{i}^{-1}}{{\sum}_{i=j^{*}}^{N} {q_{i}^{2}} s_{i}^{-1}}. $$

(39)

Consequently, the optimal value j^∗ can be computed by solving the following problem:

$$ \max_{j} \left( \sum\limits_{i=j}^{N} x_{i}^{+}(e^{*}) q_{i} e^{*} \right), $$

(40)

where e^∗ is computed relative to j, following Eq. 39. As optimal solutions verify condition (36), surviving species at MMSY display positive biomasses, and this problem is equivalent to:

$$ \max_{j} \left( \sum\limits_{i=j}^{N} \left( \frac{r_{i} - q_{i} e^{*}}{s_{i}} \right) q_{i} e^{*} \right). $$

(41)

The expression of total catches can be simplified as follows:

$$ \sum\limits_{i=j}^{N} \left( \frac{r_{i} - q_{i} e^{*}}{s_{i}} \right) q_{i} e^{*}=e^{*} \left( \sum\limits_{i=j}^{N} r_{i} q_{i} s_{i}^{-1} - e^{*} \sum\limits_{i=j}^{N} {q_{i}^{2}} s_{i}^{-1} \right) $$

(42)

By replacing the effort with its optimal expression as given in Eq. 39, we obtain:

$$ \sum\limits_{i=j}^{N} \left( \frac{r_{i} - q_{i} e^{*}}{s_{i}} \right) q_{i} e^{*}=\frac{1}{4} \frac{\left( {\sum}_{i=j}^{N} r_{i} q_{i} s_{i}^{-1} \right)^{2}}{{\sum}_{i=j}^{N} {q_{i}^{2}} s_{i}^{-1}} $$

(43)

The optimal j^∗ is thus the solution of the following optimization problem:

$$ j^{*} = \underset{j = 1 {\ldots} N}{\arg\!\max} \left( \frac{1}{4} \frac{\left( {\sum}_{i=j}^{N} r_{i} q_{i} s_{i}^{-1} \right)^{2}}{{\sum}_{i=j}^{N} {q_{i}^{2}} s_{i}^{-1}} \right). $$

(44)

1.2 A.2 Proof of Proposition 2

In this Appendix, we aim at solving the following optimization problem:

$$ \max_{e } \sum\limits_{i = 1}^{N} p_{i} x_{i}^{+}(e) q_{i} e - c e. $$

(45)

We consider that species are ranked according to their biotechnical productivities, as expressed in Eq. 6. Then, for a given effort e,²

$$ \text{if} \exists j \in \{ 1 {\ldots} N \}, \text{so that} \frac{r_{j-1}}{q_{j-1}} \leq e < \frac{r_{j}}{q_{j}}, $$

(46)

as for every i < j we have $x_{i}^{+}(e)= 0$, we deduce that

$$ \sum\limits_{i = 1}^{N} p_{i} x_{i}^{+}(e) q_{i} e - c e = \sum\limits_{i=j}^{N} p_{i} x_{i}^{+}(e) q_{i} e - \text{c e} $$

(47)

As a result, for a given effort e, total profits can be expressed as follows:

$$\begin{array}{@{}rcl@{}} \sum\limits_{i = 1}^{N} p_{i} x_{i}^{+}(e) q_{i} e -\text{c e} \!&=&\! \sum\limits_{i=k(e)}^{N} \!\!p_{i} x_{i}^{+}(e) q_{i} e - \text{c e}, \text{with} \\ k(e)\!&=&\! \min \left( \!i \!\in \!\{ 1 {\ldots} N \}, e \!< \frac{r_{i}}{q_{i}} \right). \end{array} $$

(48)

Therefore, optimization problem (45) can be written as follows:

$$ \max_{e } \sum\limits_{i = 1}^{N} p_{i} x_{i}^{+}(e) q_{i} e - \text{c e} = \max_{e} \sum\limits_{i=k(e)}^{N} p_{i} x_{i}^{+}(e) q_{i} e - \text{c e} , $$

(49)

where k(e) is defined in Eq. 48. Denote by e^∗ an optimal solution of this problem, and by k^∗ = k(e^∗) the corresponding first non-extinct species. By definition, optimal solutions verify the following condition:

$$ \frac{r_{k^{*}-1}}{q_{k^{*}-1}} \leq e^{*} < \frac{r_{k^{*}}}{q_{k^{*}}}. $$

(50)

As k^∗ is the surviving species with the lowest biotechnical productivity, species k^∗ to N are not extinct at MMEY, and their biomasses are thus larger than zero. The optimization problem (45) is then equivalent to:

$$ \max_{e} \left( {\sum}_{i=k^{*}}^{N} p_{i} \left( \frac{r_{i} - q_{i} e}{s_{i}} \right) q_{i} e - \text{c e} \right), $$

(51)

The optimal effort can be computed by using first-order optimality conditions:

$$ \frac{\partial \left( {\sum}_{i=k^{*}}^{N} p_{i} \left( \frac{r_{i} - q_{i} e}{s_{i}} \right) q_{i} e - \text{ce} \right)}{\partial e} = 0. $$

(52)

We obtain:

$$ e^{*} = \frac{1}{2} \frac{{\sum}_{i=k^{*}}^{N} r_{i} p_{i} q_{i} s_{i}^{-1} - c}{{\sum}_{i=k^{*}}^{N} p_{i} {q_{i}^{2}} s_{i}^{-1}}. $$

(53)

Consequently, the optimal value k^∗ can be computed by solving the following problem:

$$ \max_{k} \left( \sum\limits_{i=k}^{N} p_{i} x_{i}^{+}(e^{*}) q_{i} e^{*} - \text{c e}^{*} \right), $$

(54)

where e^∗ is computed relative to k, following Eq. 53. As optimal solutions verify condition (50), surviving species at MMEY display positive biomasses, and this problem is equivalent to:

$$ \max_{k} \left( \sum\limits_{i=k}^{N} p_{i} \left( \frac{r_{i} - q_{i} e^{*}}{s_{i}} \right) q_{i} e^{*} - \text{c e}^{*} \right). $$

(55)

The expression of total profits can be simplified as follows:

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{i=k}^{N} p_{i} \left( \frac{r_{i} - q_{i} e^{*}}{s_{i}} \right) q_{i} e^{*} - \text{c e}^{*}\\ &=& e^{*} \left( \sum\limits_{i=k}^{N} p_{i} r_{i} q_{i} s_{i}^{-1} - e^{*} \sum\limits_{i=k}^{N} p_{i} {q_{i}^{2}} s_{i}^{-1} - c \right) \end{array} $$

(56)

By replacing the effort with its optimal expression as given in Eq. 53, we obtain:

$$ \sum\limits_{i=k}^{N} p_{i} \left( \frac{r_{i} - q_{i} e^{*}}{s_{i}} \right) q_{i} e^{*} - \text{c e}^{*}=\frac{1}{4} \frac{\left( {\sum}_{i=k}^{N} r_{i} p_{i} q_{i} s_{i}^{-1} - c \right)^{2}}{{\sum}_{i=k}^{N} p_{i} {q_{i}^{2}} s_{i}^{-1}} $$

(57)

The optimal k^∗ is thus the solution of the following optimization problem:

$$ k^{*} = \underset{k = 1 {\ldots} N}{\arg\!\max} \left( \frac{1}{4} \frac{\left( {\sum}_{i=k}^{N} r_{i} p_{i} q_{i} s_{i}^{-1} - c \right)^{2}}{{\sum}_{i=k}^{N} p_{i} {q_{i}^{2}} s_{i}^{-1}} \right). $$

(58)

1.3 A.3 Proof of Proposition 4

Let us consider that species k has the lowest MSY effort $e_{k}^{\textsc {msy}}$. Species k is overharvested at MMSY if the following difference is positive:

$$ \sum\limits_{i=j^{*}}^{N} \alpha_{i} e_{i}^{\textsc{msy}} - e_{k}^{\textsc{msy}}=\sum\limits_{i=j^{*}}^{N} \alpha_{i} (e_{i}^{\textsc{msy}} - e_{k}^{\textsc{msy}}), $$

(59)

as ${\sum }_{i=j^{*}}^{N} \alpha _{i} = 1$. From Eq. 14, this sum is greater or equal to zero. But if at least two species differ in the sense that $e_{i}^{\textsc {msy}}<e_{j}^{\textsc {msy}}$, then this sum becomes strictly positive. Species k is then overharvested at MMSY. Using a similar proof relying on the ranking (14), it can be shown that the species with the largest MSY effort is always underharvested at MMSY.

1.4 A.4 Proof of Proposition 9

Let $p_{k}^{\prime }=p_{k}+\tau _{k}$ be the subsidized price of species k, with τ_k positive. We suppose that the subsidy does not change the number of harvested species, so that k^∗ keeps the same value. Then,

$$ e^{\textsc{mmey}^{\prime}} = \frac{1}{2}\frac{{\sum}_{i=k^{*}}^{N} r_{i}p_{i}q_{i}s_{i}^{-1} - c + r_{k}q_{k}s_{k}^{-1}\tau_{k}}{{\sum}_{i=k^{*}}^{N} p_{i}{q_{i}^{2}}s_{i}^{-1} + {q_{k}^{2}}s_{k}^{-1}\tau_{k}}=\frac{1}{2}\frac{\gamma + \delta \tau_{k}}{\rho + \phi \tau_{k}}, $$

(60)

with $\gamma ={\sum }_{i = 1}^{N} r_{i}p_{i}q_{i}s_{i}^{-1}-c$, $\delta =r_{k}q_{k}s_{k}^{-1}$, $\rho = {\sum }_{i = 1}^{N} p_{i}{q_{i}^{2}}s_{i}^{-1}$ and $\phi ={q_{k}^{2}}s_{k}^{-1}$. The effect of subsidy τ_k is given by differentiating this expression with respect to τ_k:

$$ \frac{\partial e^{\textsc{mmey}^{\prime}}}{\partial \tau_{k}}=\frac{1}{2}\frac{\delta \rho -\gamma \phi}{(\rho+ \phi \tau_{k})^{2}} $$

(61)

This derivative is negative if

$$ \frac{\gamma}{\rho} > \frac{\delta}{\phi} \Leftrightarrow \frac{{\sum}_{i = 1}^{N} r_{i}p_{i}q_{i}s_{i}^{-1}-c}{{\sum}_{i = 1}^{N} p_{i}{q_{i}^{2}}s_{i}^{-1}} > \frac{r_{k}}{q_{k}} \Leftrightarrow e^{\textsc{mmey}} > e_{k}^{\textsc{msy}}. $$

(62)

It means that if species k is overharvested at MMEY ($e^{\textsc {mmey}}>e_{k}^{\textsc {msy}}$), subsidies on species k reduce the effort at MMEY and thus improve sustainability. If the new effort becomes lower than $r_{k^{*}-1} / q_{k^{*}-1}$, then according to the demonstrations of Propositions 1 and 2, a new MMEY effort has to be recomputed that includes species k^∗− 1. Likewise, by considering that $p_{k}^{\prime }=p_{k}-\tau _{k}$ is the taxed price of species k, it can be found that taxing species that are underharvested at MMEY also reduces the effort at MMEY. Again, if the MMEY effort becomes small enough, this can increase the number of species in the system.

1.5 A.5 Proof of Proposition 10

It is equivalent to proving that a vector of prices can be found that avoids overharvesting at MMEY. From Proposition 8, we know that if $\forall k \in \left [ 1,...,N \right ], c\geq {\sum }_{i = 1}^{N} \frac {p_{i} {q_{i}^{2}}}{s_{i}} (\frac {r_{i}}{q_{i}}-\frac {r_{k}}{q_{k}})$, that is c ≥ c_sus as defined in Eq. 20, then all species are either underharvested or fully harvested at MMEY. In matrix form, it is equivalent to: $\text {MP} \leq \mathcal {C}$, with $P = \left (\begin {array}{c} p_{1} \\ {\vdots } \\ p_{N} \end {array}\right )$, $\mathcal {C} = \left (\begin {array}{c} c \\ {\vdots } \\ c \end {array}\right )$ and $M=\left (\begin {array}{cccccc} 0 & {\ldots } & \frac {{q_{N}^{2}}}{s_{N}}(\frac {r_{N}}{q_{N}}-\frac {r_{1}}{q_{1}}) \\ {\vdots } & {\ddots } & {\vdots } \\ \frac {{q_{1}^{2}}}{s_{1}}(\frac {r_{1}}{q_{1}}-\frac {r_{N}}{q_{N}}) &{\ldots } & 0 \end {array} \right )$. Following a corollary to Farkas’ lemma described in [4, 5], only one of the following alternatives holds: either ∃ P∈R^N so that $\text {MP} \leq \mathcal {C}$ and P ≥ 0, or else ∃μ ∈R^N so that μ^′M ≥ 0, $\mu ^{\prime } \mathcal {C} < 0$ and μ > 0 (μ^′ being the transpose of vector μ). As $\mathcal {C} >0$, only the first alternative holds. It is thus always possible to find a system of prices that avoids overharvesting at MMEY, by reducing the effort at MMEY. As reducing the MMEY efforts amounts to subsidizing overharvested species and taxing underharvested species (Proposition 9), it is always possible to find a system of taxes and subsidies that precludes overharvest and extinction at MMEY.

1.6 A.6 Case Study with Ecological Interactions

In this Appendix, we test the validity of the assumption of neglecting trophic interactions in the case study of the coastal fishery in French Guiana. To do this, we study the following system:

$$ x_{i}(t + 1)=x_{i}(t) \left( 1+r_{i} - \sum\limits_{j} s_{\text{ij}}x_{j}(t)-q_{i}e(t) \right), $$

(63)

where s_ij describes the interaction between species i and j. If i = j, s_ij describes intraspecific competition. If i ≠ j, s_ij is positive if i preys upon j and negative if j preys upon i. Calibrated parameters are drawn from [9]. As in the main text, we assume that the proportion of each fleet remains constant.

To compute the biomass deviation from MSY, we numerically identify efforts corresponding to all monospecific MSY, to MMSY, and to MMEY. For effort values ranging between 0 and the effort at which all species are extinct, we run the system to equilibrium, and we compute catches of all species. For all effort values, sharks and groupers disappeared in the long term. The preliminary assumption of not considering these two top predators is thus reasonable. We then identify efforts at which catches of each species are maximized, and efforts at which total catches (MMSY) and total profits (MMEY) are maximized. This allows us to compute the biomass deviation from monospecific MSY at MMSY and MMEY. Results are shown in Fig. 5. As in the main text, we only represent species that were not extinct at both MMSY and MMEY. Results are almost exactly similar to results from the main text. Very small differences exist, such as the fact that the biomass deviation at MMEY of the South American silver croaker is slightly closer to zero with trophic interactions. Thus, the assumption of neglecting trophic interactions in the case study of the coastal fishery in French Guiana does not affect our numerical results.

Fig. 5

Sustainability of MMSY and MMEY policies in the coastal fishery in French Guiana, in the presence of ecological interactions. Deviation of the harvested species’ biomasses from their MSY levels are shown. A − 100% deviation indicates that the species is extinct. As the Green weakfish, the Common snooks, the Gillbacker sea catfish, the sharks and the groupers are extinct at MMSY and MMEY, their corresponding deviations are not shown on this figure. Abbreviations are explained in Table 1

1.7 A.7 Extension to Other Growth and Production Functions

In this Appendix, we test whether our main results hold with more complex growth and production functions. We focus on MMSY results, as in these cases analytical MMEY results are difficult, if not impossible, to obtain. We first consider the case of a logistically growing population harvested with a Cobb-Douglas production function h = qx^ae^b. For the sake of a tractable analysis, we consider that a = 1. It comes that the monospecific MSY effort is

$$ e_{i}^{\textsc{msy}} = \left( \frac{r_{i}}{2q_{i}} \right)^{\frac{1}{b}}, $$

(64)

and that the MMSY effort is

$$ e^{\textsc{mmsy}} \!= \left( \!\sum\limits_{i} \alpha_{i} \left( e_{i}^{\textsc{msy}} \right)^{b} \right)^{\frac{1}{b}}, \quad \text{with} \quad \alpha_{i}=\frac{{q_{i}^{2}} s_{i}^{-1}}{{\sum}_{j = 1}^{N} {q_{j}^{2}} s_{j}^{-1}}, $$

(65)

under the assumption that no species extinction occurs at MMSY. Here, the expression of the MMSY effort is a weighted power mean of monospecific MSY efforts. As a consequence, Propositions 4 and 5 are still valid. Note that if b = 1, we find the same result as in Proposition 3.

Let us now consider a model with a Gompertz growth function and a Schaefer harvest function. The model can be written as follows:

$$ x_{i}(t + 1)=x_{i}(t) \left( 1+r_{i} ln\left( \frac{K_{i}}{x_{i}(t)} \right) -q_{i} e \right) $$

(66)

Equilibrium stocks are given by:

$$ x_{i}(e) = K_{i} \exp \left( - \frac{q_{i} e}{r_{i}} \right), $$

(67)

and the monospecific MSY effort is

$$ e_{i}^{\textsc{msy}} = \frac{r_{i}}{q_{i}} $$

(68)

The MMSY effort is then

$$ e^{\textsc{mmsy}} = \frac{1}{{\sum}_{i} \frac{\alpha_{i}}{e_{i}^{\textsc{msy}}}} \qquad \text{with} \qquad \alpha_{i} = \frac{q_{i} x_{i}(e)}{{\sum}_{i} q_{i} x_{i}(e)}, $$

(69)

under the assumption that no species extinction occurs at MMSY. Here, the MMSY effort is a weighted harmonic mean of MSY efforts. Thus, as in the previous case, Propositions 4 and 5 are still valid. Arithmetic, power, and harmonic means all belong to the generalized mean family. Therefore, these results suggest that the MMSY effort can be expressed as a weighted generalized mean, which precise form depends on the type of function considered.