SHARING THE NORTHEAST ARCTIC COD: POSSIBLE EFFECTS OF CLIMATE CHANGE

ABSTRACT. The Northeast Arctic cod inhabits the exclusive economic zones of Norway and Russia and migrates extensively between these zones. The stock is shared evenly between the two countries, with a small allocation to third countries. Higher temperatures in the Barents Sea and the Norwegian Sea are expected to affect the stock, probably increasing its size and leading to a larger share inhabiting the Russian economic zone. It is also conceivable that some spawning will begin to take place off the coast of Russia in addition to the spawning that now occurs exclusively in Norwegian waters. This paper looks at the implication of this for the division of the stock between the two countries. It is found that a greater presence of the stock in the Russian zone would strengthen rather than weaken the Norwegian bargaining position if the unit cost of fish is not sensitive to the size of the stock. If, on the other hand, the fishing costs are proportional to fishing mortality, Norway's position would be weakened almost on par with the fall in its share of the stock. The paper uses a Beverton‐Holt year class model with a Ricker recruitment function. The recruitment function is hump‐shaped, implying that a too large spawning stock is harmful for recruitment. Strong density‐dependence in the survival of eggs and larvae is a possible reason for this. It is shown that, for a stock being limited by carrying capacity at the pre‐recruit stage rather than the post‐recruit stage, one may expect a strongly asymmetric curve for sustainable yield as a function of total biomass. The biomass of an exploited population might possibly exceed the biomass of a pristine population under those circumstances.     

DO ACCURATE STOCK ESTIMATES INCREASE HARVEST AND REDUCE VARIABILITY IN FISHERIES YIELDS?

Abstract Fisheries managers normally make decisions based on stock abundance estimates subject to process, observation, and model uncertainties. Considerable effort is invested in gathering information about stock size to decrease these uncertainties. However, few studies have evaluated benefits from collecting such information in terms of yield and stability of annual harvest. Here, we develop a strategic age‐structured population model for a long‐lived fish with stochastic recruitment, resembling the Norwegian spring‐spawning herring (NSSH, Clupea harengus L.). We evaluate how uncertainties in population estimates influence annual yield, spawning stock biomass (SSB), and variation in annual harvest, using both the proportional threshold harvesting (PTH) and the current harvest control rule for NSSH as harvest strategies. Results show that the consequences of a biased estimate are sensitive to the harvest strategy employed. If the harvest strategy is suitably chosen, the benefits of accurate information are low, and less information about the stock is necessary to maintain high average yield. Reduced harvest intensity effectively removes the need for accurate stock estimates. PTH (a variant of the constant escapement strategy) with low harvest ratio and the current NSSH harvest control rule both provide remarkable stability in yield and SSB. However, decreased uncertainty will often decrease year‐to‐year variation in harvest and the frequency of fishing moratoria.     

ON THE OPTIMAL POLICY FOR COMBINING HARVESTING OF PREDATOR AND PREY

An optimal policy is sought for maximizing present value from the combined harvest of two ecologically dependent species, which would coexist as predator and prey in the absence of harvesting. Harvest rate for each species is assumed proportional to both stock level and effort. For a large class of biological productivity functions, it is established that the optimal equilibrium point in the phase-plane of stock levels must be a saddle-point. For quadratic productivity functions, a combination of analytical reasoning and numerical experiment is used to show that first and second order necessary conditions for optimality are satisfied by a unique approach to equilibrium, which must therefore be optimal. The corresponding control law is given, and an apparent suggestion of two previous authors concerning the policy is shown to be inappropriate. The optimal policy enables an estimate to be made of the true loss of resource value due to a catastrophic fall in stock level.     

EQUILIBRIUM ABUNDANCE OF SALMON STOCKS IN A LIFE CYCLE MODEL WITH INTERACTING HATCHERY AND NATURAL SUBSTOCKS

This paper develops the equilibrium solutions for an age-structured life cycle model where spawning stock is split between natural and hatchery spawners. Mixing is allowed between the stocks through natural stock take by the hatchery and release of eggs or fry by the hatchery when its capacity is exceeded. The natural stock is assumed to have density-dependent egg-smolt survival while the hatchery stock has linear survival. The model can be applied to any hatchery reared fish stock but is most appropriate for salmon, where hatchery and naturally spawned fish mix completely later in life. Questions about the mix between the hatchery and natural stocks can be addressed by computing the fraction of naturally and hatchery derived stock among the natural and hatchery spawners as well as among the total adult run. Columbia River chinook stock are used as an example for which equilibria and mixing fractions are computed. A Monte Carlo sensitivity study on model parameters showed that the natural stock survival from smoltification to age 1 and the natural basin smolt carrying capacity are most important in controlling the equilibrium age-1 naturally spawned stock. Changing hatchery capacity over two orders of magnitude showed a 50 percent change in the fraction of naturally derived fish in the natural spawning stock, while the relative size of natural and hatchery stocks changed over two orders of magnitude. The model can serve as a tool for quickly assessing the effects of spawning habitat modification and hatchery supplementation practices on long-term stock mixing and stock abundance.     

OPTIMAL DISCOUNTING OF BENEFITS FROM CLEANUP AT WASTE SITES

Abstract This paper uses a general equilibrium optimal growth model to discuss the role of optimal discounting of future benefits from cleanup at high‐level toxic waste sites. Cleanup simultaneously generates two streams of benefits. One of these is directly from utility and the other is indirectly from the added productivity of workers. We note that the optimal discount rate is different for these two types of benefits. Along the optimal path, the former are discounted at the rate of time preference and the latter at the market rate of interest. We achieve this by identifying four components of the shadow value of the stock of toxic waste. These are the utility, productivity, cost, and abundance effects. The distinction between discount rates appears to have been overlooked in the literature but has significant implications for environmental cost‐benefit analysis due to the growing interest in applying zero time preference to environmental problems, (like waste cleanup) whose consequences extend many generations into the future. A numerical example is included to illustrate these concepts.     

REBUILDING THE EASTERN BALTIC COD STOCK UNDER ENVIRONMENTAL CHANGE (PART II): TAKING INTO ACCOUNT THE COSTS OF A MARINE PROTECTED AREA

Abstract This study adds a cost analysis of the Eastern Baltic cod fishery to the existing model presented in Röckmann et al. [2007a] . As cost data on this international fishery do not exist, data from Denmark are extrapolated to the whole international fishery. Additionally, unit and total variable costs are simulated, and the sensitivity to a set of different cost–stock and cost–output elasticities is tested. The study supports preliminary conclusions that a temporary marine reserve policy, which focuses on protecting the Eastern Baltic cod spawning stock in the International Council for the Exploration of the Sea (ICES) subdivision 25, is a valuable fisheries management tool to (i) rebuild the overexploited Eastern Baltic cod stock and (ii) increase operating profits. The negative effects of climate change can be postponed for at least 20 years—depending on the assumed rate of future climate change. Including costs in the economic analysis does not change the ranking of management policies as proposed in the previous study where costs were neglected.     

REBUILDING THE EASTERN BALTIC COD STOCK UNDER ENVIRONMENTAL CHANGE–A PRELIMINARY APPROACH USING STOCK,ENVIRONMENTAL, AND MANAGEMENT CONSTRAINTS

ABSTRACT. . The population dynamics of the Eastern Baltic cod (Gadus morhua callarias L.), unlike many other stocks, shows a strong dependency on environmental conditions. To test the implications of different management policies on the stock and the fishery in a system of global environmental change, we apply a spatially disaggregated, discrete time, age‐structured model of the Eastern Baltic cod stock in 50 year simulation analyses. The simulation provides an analysis of stock, yield, and revenue development under various management policies and environmental scenarios. The policy analysis, focusing on different regulations of fishing mortality, is embedded into three environmental scenarios, assuming low, medium, or high climate and environmental change. The environmental assumptions are based on simulation results from a coupled atmosphere‐ocean regional climate model, which project salinity in the Baltic Sea to decrease by 7–47% in the period 2071–2100 relative to the reference period 1961 1990. Our simulation results show that a significant reduction in fishing mortality is necessary for achieving high long‐term economic yields. Moreover, under the environmental scenarios presented, a stock collapse cannot be prevented. It can, however, be postponed by the establishment of a marine reserve in ICES subdivision 25.     

THE EFFECTS OF DIFFERENT STRATEGIC VARIABLES IN NONCOOPERATIVE FISHERIES GAMES

Abstract In this paper, we use stock size, harvest quantity, and fishing effort as strategic variables. We model a two‐agent noncooperative fishery game, where the agents (nations) harvest a common fish stock. The planning horizon is infinite. The model is solved successively using one instrument at a time as the strategic variable in the game. The net present values of fishing and the escapement stock level from the three different models are compared to show how the choice of variables affects the results. The choice of strategic variable is not a trivial one, as the results are shown to be sensitive to the discounting, the stock's rate of growth, and the assumptions about the distribution of the fish in response to harvesting.     

OPTIMAL ENVIRONMENTAL TAXES: EFFECTS OF POLLUTION DECAY AND CONSUMER AWARENESS

Abstract The effects of nonlinear decay and consumer preferences are analyzed in a setting where optimal extraction of nonrenewable resources is combined with stock externalities. The control is exercised via a corrective tax and the time horizon is divided into two periods: an initial phase with extraction and a terminal phase without extraction. The time horizon with extraction is determined endogenously. The model does not assume separability of the objective function. The purpose here is to demonstrate that relatively simple deviations from the standard assumptions, such as linear decay and no consumer awareness, may have large effects. Sensitivity analyses indicate large differences in the optimal extraction period, the total level of extraction and cumulative emissions depending on the form of the decay function and the presence of consumers' awareness for the environment.     

FISHERY BENEFITS OF FULLY PROTECTED MARINE RESERVES: WHY HABITAT AND BEHAVIOR ARE IMPORTANT

ABSTRACT. Fully protected marine reserves, areas that are closed to all fishing, have attracted great interest for their potential to benefit fisheries. A wide range of models suggest reserves will be most effective for species that are relatively sedentary as adults but produce offspring that disperse widely. Adult spawning stocks will be secure from capture in reserves, while their offspring disperse freely into fishing grounds. Such species include animals like reef fish, mollusks and echino‐derms, and models typically indicate that when they are over‐fished, catches will be higher with reserves than without. By contrast, the same models suggest that reserves will be ineffective for animals that are mobile as adults species like cod, tuna or sharks. They remain vulnerable to fishing whenever they move outside reserves. Unfortunately, most models lack sufficient realism to effectively gauge reserve effects on migratory species. They usually assume that individuals are homogeneously distributed in a uniform sea and move randomly. They also assume that fishers hunt at random. Neither is true. For centuries, fishers have targeted places and times when their quarry are most vulnerable to capture. Protecting these sites could have disproportionately large effects on stocks. Furthermore, models rarely take into account possible benefits from improvements in habitat within reserves. Such changes, like increased biomass and complexity of bottom‐living organisms, could alter fish movement patterns and reduce natural mortality rates in ways that enhance reserve benefits. We present a simple model of reserve effects on a migratory fish species. The model incorporates spatial variation in vulnerability to capture and shows that strategically placed reserves can offer benefits in the form of increased spawning stock and catch, especially when fishing intensities are high. We need to develop a new generation of models that incorporate habitat and behaviour to better explore the utility of reserves for mobile species. Migratory behavior does not preclude reserves from benefiting a species, but it demands that we apply different principles in designing them. We must identify critical sites to species and develop reserve networks that focus protection on those places.     

Optimal Control of a Logarithmic Advertising Model

A logarithmic advertising model is posed and solved for an optimal dynamic advertising policy for both finite and infinite horizon cases by using optimal control theory. In the case of the infinite horizon, the optimal long-run stationary equilibrium rate of advertising is obtained. It is shown that the optimal advertising policy is independent of the initial level of sales for the logarithmic model.     

FROM THE GOLDEN AGE TO THE NEW INDUSTRIAL AGE: FISHERY MODELING IN THE EARLY 21ST CENTURY

ABSTRACT. If the Golden Age of fishery modeling is over, the New Industrial Age is beginning. In this new age (the early decades of the 21st century), we¹ expect to see increasing use of high‐level modeling tools, assessments closely tailored to each stock and its data, more multi‐agency collaboration, wider use of multiple‐model assessments, and extensive peer review of stock assessments. We hope that fishery management, as well, will be seen as a proper object of peer review, from which it should benefit substantially. We believe that increasing attention will be paid to financial efficiency, both in shaping data collection programs and in specifying what information must be modeled to effect management. If presently depleted stocks start to recover, analysts will be required to examine in more detail the dynamics of increasing stocks. That would constitute a pleasant duty, and one that may yield new insights in population biology.     

Optimal asset allocation for aggregated defined benefit pension funds with stochastic interest rates

In this paper we study the optimal management of an aggregated pension fund of defined benefit type, in the presence of a stochastic interest rate. We suppose that the sponsor can invest in a savings account, in a risky stock and in a bond with the aim of minimizing deviations of the unfunded actuarial liability from zero along a finite time horizon. We solve the problem by means of optimal stochastic control techniques and analyze the influence on the optimal solution of some of the parameters involved in the model.     

Groundwater management: Waiting for a drought*

In this article we present a stylized model for optimal management of an unconfined groundwater resource when the threat of drought exists. The drought is modeled as a stochastic event that hits at an uncertain date and two benchmark management policies are investigated: (a) A policy of optimal dynamic management ignoring the threat of drought; and (b) an economically optimal policy that accounts for the threat of a drought. We show that the optimal predrought steady‐state equilibrium stock size of groundwater under policy b is larger than that under policy (a) Furthermore, we show that an increase in the probability of a drought gives rise to two counteracting effects: One in the direction of a larger predrought steady‐state equilibrium stock size (a recovery effect) and one in the direction of a lower predrought steady‐state equilibrium stock (an extinction effect). We find that the recovery effect dominates the extinction effect. Recommendations for Resource Managers: We analyze two groundwater extraction policies that can be used when a threat of drought exists: (a) Dynamic optimal management ignoring the threat of drought; and (b) dynamic optimal management taking the threat of drought into account. We show that the predrought steady‐state equilibrium stock size of water should be larger under the policy (b) than under policy (a). This conclusion has three implications for resource managers:

• Current groundwater management should take future extraction possibilities into account.

• A resource manager ought to take the threat of drought into account in groundwater management.

• A buffer stock of water should be built‐up before the drought to be drawn upon during the event.

     

A STOCHASTIC FEEDBACK MODEL FOR OPTIMAL MANAGEMENT OF RENEWABLE RESOURCES

Analytical expressions for optimal harvest of a renewable resource stock which is subject to a stochastic process are found. These expressions give the optimal harvest as an explicit feedback control law. All relations in the model, including the stochastic process, may be arbitrary functions of the state variable (stock). The objective function, however, is at most a quadratic function in the control variable (yield). A quadratic objective function includes the cases of downward sloping demand and increasing marginal costs which are the most common sources for nonlinearities in the economic part of the model. When it is assumed that there is a moratorium on harvest for stock sizes below a certain level (biological barrier), it is shown that the barrier requirements influence the optimal harvest paths throughout.     

THE EFFECTS OF ITQ IMPLEMENTATION: A DYNAMIC APPROACH

ABSTRACT. This paper investigates the intertemporal effects of introducing Individual Transferable Quota, ITQ, fishery management programs on stock size, fleet size and composition, and returns to quota holders and to vessel operators. Theoretical analysis is conducted using a specific version of a general dynamic model of a regulated fishery. It is demonstrated that the effects will differ depending upon the prevailing regulation program, current stock size, and existing fleet size, composition and mobility and upon how the stock and fleet change over time after the switch to ITQs. The paper expands upon previous works by modeling the dynamics of change in fleet and stock size and by allowing for changes in the TAC as stock size changes, by comparing ITQs to different regulations, and by allowing the status quo before ITQ implementation to be something other than a bioeconomic equilibrium. Specific cases are analyzed using a simulation model. The analysis shows that the annual return per unit harvest to quota owners can increase or decrease over the transition period due to counteracting effects of changes in stock and fleet size. With ITQs denominated as a percentage of the TAC, the current annual value of a quota share depends upon the annual return per unit of harvest and the annual amount of harvest rights. Because the per unit value can increase or decrease over time, it is also possible that the total value can do the same. Distribution effects are also studied and it is shown that while the gains from quota share received are the present value of a potentially infinite stream of returns, potential losses are the present value of a finite stream, the length of which depends upon the remaining life of the vessel and the expected time it will continue to operate.     

A kind of non‐traditional biomanipulation model with constant releasing fish

On the basis of the ideas of non‐traditional biomanipulation control in fresh water body, a kind of nutrient–algae fish model is presented to investigate the effects of constant releasing fish on the nutrient and the algae. The threshold conditions for the extinction of the algae are obtained by discussing the stability of boundary equilibrium. The conditions for the coexistence of the algae and the fish are obtained by discussing the existence and stability of positive equilibrium. Besides, Hopf bifurcation is also analyzed by considering the parameter about the amount of the released fish. Furthermore, a kind of optimal problem is presented, and the necessary condition for the existence of the optimal solution is given by Pontryagin maximum principle. Finally, the mathematical results are verified by numerical simulations. The mathematical results show that there is a threshold amount of the released fish, above which the activity of releasing fish can control the growth of the algae and further reduce the probability of the algae bloom, but can not decrease the eutrophication level of the water body. Copyright © 2017 John Wiley & Sons, Ltd.     

HARVESTING ECONOMIC MODELS AND CATCH‐TO‐BIOMASS DEPENDENCE: THE CASE OF SMALL PELAGIC FISH

Abstract In the case of small pelagic fish, it seems reasonable to consider harvest functions depending nonlinearly on fishing effort and fish stock. Indeed, empirical evidence about these fish species suggests that marginal catch does not necessarily react in a linear way neither to changes in fishing effort nor in fish stock levels. This is in contradiction with traditional fishery economic models where catch‐to‐input marginal productivities are normally assumed to be constant. While allowing for nonlinearities in both catch‐to‐effort and catch‐to‐stock parameters, this paper extends the traditional single‐stock harvesting economic model by focusing on the dependence of the stationary solutions upon the nonlinear catch‐to‐stock parameter. Thus, we analyze equilibrium responses to changes in this parameter, which in turn may be triggered either by climatic or technological change. Given the focus in this study on the case of small pelagic fish, the analysis considers positive but small values for the catch‐to‐stock parameter.     

Stock repurchase with an adaptive reservation price: A study of the greedy policy

We consider the problem of stock repurchase over a finite time horizon. We assume that a firm has a reservation price for the stock, which is the highest price that the firm is willing to pay to repurchase its own stock. We characterize the optimal policy for the trader to maximize the total number of shares that they can buy over a fixed time horizon. In particular, we study a greedy policy, which involves in each period buying a quantity that drives stock price to the reservation price.     

一类可再生资源系统的最优动态平衡收获

研究一类可再生资源系统的最优利用问题．首先,引进一个新的效用函数, 它依赖于收获努力度和资源量,由此导出最优控制问题．其次证明该控制问题最优解的存在性．然后,利用无穷区间上控制问题的最大值原理,得到一个非线性的四维最优系统．通过对上述系统正平衡解的详细分析,借助 Hopf 分支定理证明了极限环的存在性．之后考虑中心流形上的简化系统, 分析极限环的稳定性．最后,解释所得结果的生物经济学意义．