Ecological Archives A016-056-A1

James N. Sanchirico, Urmila Malvadkar, Alan Hastings, and James E. Wilen. 2006. When are no-take zones an economically optimal fishery management strategy? Ecological Applications 16:1643–1659.

Appendix A. Solution techniques, optimal biomass densities, and dispersal flows.

Our model is a two-state, two control optimal control problem that is linear in the controls (catch rates). The dynamics are "bang-bang-singular" in nature, which means that the control is potentially set at either of its bounds (upper and lower) for a period of time, or at an interior (singular) control. In a one patch setting, a bang-bang-singular solution implies, for example, that catch is set to its maximum value (when the fish population starts above its steady-state solution) for a period of time (Clark 1990; Kamien and Schwartz 1991). As the population level is reduced, the control will switch off of its maximum value at some point. Whether the catch next goes to the lower bound (zero) or to its singular level depends on the characteristics of the problem. In the former case, if the catch switches from its maximum to zero (minimum value), this will allow the stock to recover (i.e., temporary moratorium on fishing). Generally, the final switching point occurs when the control switches back to its singular value, which will or will not vary over time depending on the nature of the control problem.

To solve for the optimal catch levels over time, the regulator maximizes the objective function subject to the population growth functions (Eqs. 1 and 2 in the main body of the paper), and a set of initial conditions. A reasonable initial condition would place the two populations (x1,x2) at their open-access equilibria, which reflects unregulated conditions. Open-access refers to an equilibrium where profits are driven to zero, in this case when (x1oa,x2oa) = (c1/p1, c2/p2).

The optimal catch and population level can be determined by choosing the catch levels, hi(t), in order to maximize the Hamiltonian, which is defined as:

where i are the shadow prices or adjoint variables that represent the marginal value of an additional unit of patch x1 or x2 biomass on the present discounted fishery profits (Kamien and Schwartz 1991).

Since the problem is linear in the controls, we can first rearrange the Hamiltonian to isolate the control variables. Once this is done, we observe that there are switching functions in the Hamiltonian that we designate as:

These switching functions are the time-varying coefficients that multiply each of the controls in the rearranged Hamiltonian. By the Pontryagin necessary conditions, each control must be chosen to maximize the Hamiltonian at each instant. Since controls enter the Hamiltonian linearly, the optimal levels of the control instruments must satisfy:

When a switching function is positive, the optimal control for that patch is set at its maximum and when the switching function is negative the control must be set at its minimum allowable value. If the switching function is zero, the control must be set at its "singular value", which is to be determined as discussed below. In addition to the switching functions, the necessary conditions include the biomass state equations (Eqs. 1 and 2 in the main body of the paper) along with the adjoint equations. With respect to the latter, Pontryagin's Principle states that for an optimal solution (x1*, x2*, h1*, h2*) there exist adjoints 1 and 2 such that
      (A.1)


Double Singular Solutions

While bang-bang control solutions consist of periods where the harvest may be zero, a maximum value, or a "singular" value in between, the complexity of even this simple model prohibits us from analyzing all of the possible permutations. Instead we assume that both of the controls (harvest rates) are at their (interior) singular values. This is known as the doubly-singular solution (Clark 1990).

For this optimization system, the doubly-singular solution is a steady state. Once the system reaches these values, the population size and optimal harvest in both patches are optimally held constant. We do not assess the stability properties of the equilibrium, as it is not possible analytically and we simply assume that an equilbrium exists and it is reachable. The full optimal solution may consist of a complex harvesting regime along the approach to the equilibrium, along with the equilibrium itself. We focus our analysis on the steady-state part of the solution, or the doubly-singular constant solution. We next determine the doubly-singular solution using Pontryagin's equations. Since H is linear in hi, the doubly-singular solution yields
    (A.2)

Differentiating A.2 yields:
    (A.3)

and from Pontyagin's Principle in Eq. A.1 we find:
    (A.4)

Substituting A.2 into A.1 and simplifying we have for each patch i = 1,2 with i j:
    (A.5)

These two equations implicitly specify the equilibrium (x1*, x2*) for the doubly-singular control problem, which depends on biological growth and dispersal parameters as well as the economic parameters.




Optimal densities in closed-ecological system

If the patches are independent (no dispersal processes), we find that the optimal solution involves moving to a steady-state in which the biomass in each patch satisfies:
    (A.6)
In this case, the optimal equilibrium biomass densities are functions only of own-patch specific economic and biological parameters. We also know that the optimal biomass levels could be above or below the maximum sustainable yield (MSYi = 1/2) in our closed biological system depending upon the degree of stock dependence in harvesting costs. For example, with no stock dependent costs, it can be shown that the optimal population level must satisfy the condition that Fi(xi)/ xi = , which implies that optimal population level is below MSY. In this case, it is easy to show that higher discount rates lead to lower optimal stock sizes. With stock dependent costs, this condition is more complex but Clark (1990) has shown that it may with high costs, lead to an equilibrium above MSY.




Fish directional flows and spillover benefits

In addition to the assumptions regarding the dispersal rates, the classification of patches as net exporters or importers also depends on the level of fishing in each patch, because fishing affects the relative densities of fish in the two populations. Therefore, for each set of optimal fish densities, it is possible to calculate the direction and level of the flows in the system at the optimal spatial catch rates. For example, in a system with only adult dispersal, the direction and level at the optimal solution can be determined by plugging in the optimal density levels into the net dispersal term. Recall for patch 1, the net dispersal term at the optimal solution is d12x2*-d11x1*. If this term is positive, the flow is from patch 2 to patch 1, and the level is just equal to the difference between the inflow and outflow.

To help understand the spatial fish dynamics when no-take zones are optimal solutions, we illustrate the dispersal in the system for each triplet (c1crit,x1*,x2*). These graphs illustrate the flows of dispersal as you move off of the two extreme cases (patch 1 a source or patch 1 a sink) in Fig. 3 or Fig. 5.

Fig. A1 illustrates the net ``adult'' dispersal flows (measured as a percentage of the aggregate biomass) into patch 1 at the critical set, where a positive (negative) number implies that patch 1 is on net receiving (providing) biomass from (into) patch 2. Because we allow for different dispersal rates dij, net adult dispersal is a function of these differences and the patch optimal population levels at the critical cost levels. As we move off either axis, the degree of net dispersal (mixing) depends on the relative optimal density levels and dispersal rates. For all the points to the left of the ``net dispersal equal to zero'' line patch 1 acts as a net importer (sink) and to the right a net exporter (source). The spillover or network effect from patch 1 to patch 2 increases moving from the northwest corner to the southeast corner (direction of arrow).

Comparing Fig. A1 and Fig. 3, it is easy to see that as the population dispersing into the reserve increases, the magnitude of the critical cost coefficient decreases. This implies that, as the dispersal flow increases, the dispersal value of the reserve (patch 1) to the system outweighs the local returns from fishing in the patch, because without the connectivity, it would be optimal to fish in the patch.

Fig. A2 plots the net dispersal for the case when there is only juvenile dispersal, and again solving for the critical cost coefficient. In this case, net dispersal at the optimal solution is equal to b1x1* - b2x2*, and is measured as a percentage of the aggregate optimal biomass at the critical cost levels. Relative to the adult dispersal case (Fig. A1), the dispersal flows are greater in this system, but the critical cost parameter range is smaller, everything else equal (compare the ranges of the colorbar across Fig. 3 and Fig. 5). This is due to the density-dependence in juvenile dispersal.


Figures
   FIG. A1. Net dispersal flows at the critical cost level with density dependent settlement. Note: On both axes, we measure the dispersal rate as a proportion of the patch intrinsic growth rate. The y-axis corresponds to cases where the patch 1 is a pure sink and the x-axis corresponds to the case where the patch 1 is a pure source. The contour lines correspond to net dispersal levels measured as a proportion of the aggregate biomass at the critical cost levels illustrated in Fig. 3. The zero net dispersal line bifurcates the graph, where areas to the right of this line patch 1 is a net exporter of biomass and to the left patch 1 is a net importer in the system. The spillover or network effect from patch 1 to patch 2 increases moving from the northwest corner to the southeast corner (direction of arrow).

 

 
   FIG. A2. Net dispersal flows at the critical cost level for density-independent settlement. Note: On both axes, we measure the share of juveniles that move from one patch to the other as a proportion of the patch intrinsic growth rates (recall we impose the adding up restriction on juvenile dispersal process). The contour lines correspond to net dispersal levels measured as a proportion of the aggregate biomass at the critical cost levels illustrated in Fig. 5. The zero net dispersal line bifurcates the graph, where areas to the right of this line patch 1 is a net exporter and to the left the patch 1 is a net importer in the system. Relative to Fig. A1, the dispersal flows as a proportion of growth rates are larger. The spillover or network effect from patch 1 to patch 2 increases moving from the northwest corner to the southeast corner (direction of arrow).

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