Bioeconomic Modelling of Coastal Cod and Kelp Forest Interactions: Co-benefits of Habitat Services, Fisheries and Carbon Sinks

Abstract

Ecosystem-based fisheries management seeks to expand upon the traditional one-stock fisheries management measures by internalizing the effects of fishing on marine ecosystems, and accounting for biological interactions among marine resources. The fact that marine resources provide multiple, often competing benefits, makes the accomplishment of these ecosystem-based fisheries management objectives highly complex. In this paper, we develop a dynamic bioeconomic model to analyze the ecological and economic interactions between fisheries and renewable habitat where the habitat provides multiple ecosystem services. Specifically, a single resource manager seeks to maximize co-benefits of fishery-habitat interactions when the habitat is an exploitable marine resource, but also a dwelling place for commercial fish, enhancing the growth of the fish stock and providing regulating ecosystem services in the form of carbon sink for climate change mitigation. The optimal management rules for both fishery and habitat are derived and discussed. We also present an application of the model to analyze an integrated management of coastal cod and kelp forests in Norway, where regulations on commercial harvesting of kelp forests seek to protect fisheries. Both the theoretical model and the Norwegian application suggest substantial potential increases for both coastal cod and kelp forest stocks, with an attendant 8% increase in cod harvests, and about 1% reduction in kelp harvests. In addition, an optimal management regime that internalizes carbon sink co-benefits of kelp forests stores additional 300,000 tonnes of carbon.

Appendix 1: Shape of x*(y), y*(x) and y**(x) Curves

We derive the shapes of the optimal stock curves by taking the total derivatives of Eqs. (13) and (14). First, we differentiate Eq. (13) giving:

$$\frac{{dx^{*} }}{dy} = - \frac{{\left( {p_{1} - c_{1} \left( x \right)} \right)F_{xy} \left( {x,y} \right) + c_{1x} \left( x \right)F_{y} \left( {x,y} \right)}}{{\left( {p_{1} - c_{1} \left( x \right)} \right)F_{xx} \left( {x,y} \right) + c_{1xx} \left( x \right)F\left( {x,y} \right) + \delta c_{1x} \left( x \right)}}$$

(22)

Given that $p_{1} - c_{1} \left( x \right) > 0$, $F_{xy} \left( {x,y} \right) > 0$, $F_{y} \left( {x,y} \right) > 0$ but $c_{1x} \left( x \right) < 0$, the numerator can assume any sign. In addition, since $F_{xx} \left( {x,y} \right) < 0$ but $c_{1xx} \left( x \right) > 0$, the denominator can also assume any sign. Because of the negative sign on the RHS of Eq. (22), $\frac{{dx^{*} }}{dy}$ is positive if and only if the numerator and denominator have opposite signs of each other. Starting with the case in which the denominator is positive, that is, $\left| {c_{1xx} \left( x \right)F\left( {x,y} \right)} \right| > \left| {\left( {p_{1} - c_{1} \left( x \right)} \right)F_{xx} \left( {x,y} \right) + \delta c_{1x} \left( x \right)} \right|$. Then for $\frac{{dx^{*} }}{dy} > 0$, the numerator must be negative. This requires that $\left| {c_{1x} \left( x \right)F_{y} \left( {x,y} \right)} \right| > \left| {\left( {p_{1} - c_{1} \left( x \right)} \right)F_{xy} \left( {x,y} \right)} \right|$ and $\left| {c_{1x} \left( x \right)F_{y} \left( {x,y} \right)} \right| > \left| {\left( {p_{1} - c_{1} \left( x \right)} \right)F_{xy} \left( {x,y} \right)} \right|$. Alternatively, if the numerator is positive, i.e.$\left| {\left( {p_{1} - c_{1} \left( x \right)} \right)F_{xy} \left( {x,y} \right)} \right| > \left| {c_{1x} \left( x \right)F_{y} \left( {x,y} \right)} \right|$, then $\frac{{dx^{*} }}{dy} > 0$ implies that $\left| {\left( {p_{1} - c_{1} \left( x \right)} \right)F_{xx} \left( {x,y} \right) + \delta c_{1x} \left( x \right)} \right| > \left| {c_{1xx} \left( x \right)F\left( {x,y} \right)} \right|$.

Secondly, we differentiate Eq. (14) as:

$$\frac{{dy^{*} }}{dx} = - \frac{{\left( {p_{1} - c_{1} \left( x \right)} \right)F_{yx} \left( {x,y} \right) - c_{1x} \left( x \right)F_{y} \left( {x,y} \right)}}{{\left( {p_{2} - c_{2} } \right)G_{yy} \left( y \right) + \left( {p_{1} - c_{1} \left( x \right)} \right)F_{yy} \left( {x,y} \right)}}$$

(23)

Since $p_{1} - c_{1} \left( x \right) > 0$, $F_{xy} \left( {x,y} \right) = F_{yx} \left( {x,y} \right) > 0$, $c_{1x} \left( x \right) < 0$ but $F_{y} \left( {x,y} \right)$ is indeterminate, it means that the numerator is indeterminate. However, for the denominator, $G_{yy} \left( y \right) < 0$, $F_{yy} \left( y \right) < 0$ and both $p_{1} - c_{1} \left( x \right) > 0$ and $p_{2} - c_{2} > 0$, we can conclude that the denominator is strictly negative. With the negative denominator in conjunction with the negative sign makes it positive. Therefore, for $\frac{{dy^{*} }}{dx}$ to be positive, then numerator must be positive. The numerator is only positive under three conditions: (1) $F_{y} \left( {x,y} \right) = 0$, (2) $F_{y} \left( {x,y} \right) > 0$ and (3) $F_{y} \left( {x,y} \right) < 0$ but $\left| {\left( {p_{1} - c_{1} \left( x \right)} \right)F_{yx} \left( {x,y} \right)} \right| > \left| {c_{1x} \left( x \right)F_{y} \left( {x,y} \right)} \right|$.

Finally, we differentiate Eq. (20) as:

$$\frac{{dy^{**} }}{dx} = - \frac{{\left( {p_{1} - c_{1} \left( x \right)} \right)F_{yx} \left( {x,y} \right) - c_{1x} \left( x \right)F_{y} \left( {x,y} \right)}}{{\left( {p_{2} - c_{2} } \right)G_{yy} \left( y \right) + \left( {p_{1} - c_{1} \left( x \right)} \right)F_{yy} \left( {x,y} \right) + p_{v} v_{yy} \left( y \right)}}$$

(24)

Since $v_{yy} \left( y \right) < 0$, it means that $\frac{{dy^{**} }}{dx}$ will be positive if the three conditions stated above for $\frac{{dy^{*} }}{dx}$ to be positive hold.