An infinite horizon mathematical programming model of a multicohort single species fishery

Fishery policy evaluation should take account of the initial state of the fishery and the population dynamics of the fish stock. Although multicohort bioeconomic fishery policy evaluation models have been developed, the results from these models depend on the choice of planning period and the desired state of the stock at the end of this period. In this paper it is noted that these limitations can be overcome by evaluating fishery policy over an infinite time horizon, and a mixed integer programming (MIP) model is developed for carrying out this form of analysis in a multicohort single species fishery. This new MIP model allows policies to be evaluated over an infinite horizon by incorporating results from a steady state fishery model into a multiperiod framework. The use of this MIP model in determining policies for reaching and maintaining a steady state is illustrated.     

Optimal and feedback strategies for managing multicohort populations

A Beverton and Holt type linear cohort dynamics model is integrated and combined with a nonlinear stock-recruitment relationship to obtain a discrete-time multicohort harvesting model. Assuming that each age class is individually controllable, it is shown, subject to certain assumptions, that the optimal harvesting strategy is to drive the population to the maximum sustainable yield solution in one time step. In most fisheries, this controllability assumption is not met and harvesting is agewise nonselective. In this case, it may be preferable to implement a harvesting policy based on suboptimal constant effort or stock level feedback strategies, rather than implement a more complicated optimal policy. This question is addressed through numerical studies on the management of an anchovy fishery.Dedicated to G. LeitmannThe author would like to thank M. Mangel, W. Reed, P. Sullivan, and G. Swartzman for commenting on a draft of this paper.     

HARVESTING AN AGE‐STRUCTURED POPULATION AS BIOMASS: DOES IT WORK?

Abstract The economics of fisheries is based heavily on describing fish populations by the surplus production model. Both economists and ecologists have different opinions on whether this approach provides an adequate biological basis for economic analysis. This study takes an age‐structured population model and shows how, under equilibrium conditions, it determines the surplus production model. The surplus production model is then used to solve an optimal feedback policy for a generic optimal harvesting problem. Next, it is assumed that the fishery manager applies this feedback policy even though the fish population actually evolves according to the age‐structured model. This framework is applied to the widow rockfish, Atlantic menhaden, and Pacific halibut fisheries. Population age‐structure contains information on future harvest possibilities. The surplus production model neglects this information and may lead to major deviations between the expected and actual outcomes especially under multiple steady states and nonlinearities.     

Maximum sustainable yield harvesting in an age‐structured fishery population model

A generic type age‐structured fishery population model consisting of two harvestable age classes is formulated. Optimal harvest rates are determined with uniform fishing mortality and perfectly selective fishing, respectively. Selectivity allows for differentiating the fishing mortality among different age classes. Sustainable yield–biomass functions are developed, and the maximum sustainable yield (MSY) solutions are found under both exploitation schemes. The gain of perfectly selective fishing over uniform (or biomass) fishing is examined under various assumptions, and it is proved that the benefit of selective harvesting increases when the harvestable fish population becomes more heterogeneous in terms of weights, or values. In contrast to the surplus production model, or Clark model, the analysis also demonstrates that MSY with different age classes is not purely a biological concept.     

SUSTAINABLE YIELDS IN FISHERIES: UNCERTAINTY,RISK‐AVERSION,AND MEAN‐VARIANCE ANALYSIS

Abstract We consider a model of a fishery in which the dynamics of the unharvested fish population are given by the stochastic logistic growth equation Similar to the classical deterministic analogon, we assume that the fishery harvests the fish population following a constant effort strategy. In the first step, we derive the effort level that leads to maximum expected sustainable yield, which is understood as the expectation of the equilibrium distribution of the stochastic dynamics. This replaces the nonzero fixed point in the classical deterministic setup. In the second step, we assume that the fishery is risk averse and that there is a tradeoff between expected sustainable yield and uncertainty measured in terms of the variance of the equilibrium distribution. We derive the optimal constant effort harvesting strategy for this problem. In the final step, we consider an approach that we call the mean‐variance analysis to sustainable fisheries. Similar as in the now classical mean‐variance analysis in finance, going back to Markowitz [1952] , we study the problem of maximizing expected sustainable yields under variance constraints, and with this, minimizing the variance, e.g., risk, under guaranteed minimum expected sustainable yields. We derive explicit formulas for the optimal fishing effort in all four problems considered and study the effects of uncertainty, risk aversion, and mean reversion speed on fishing efforts.     

BENEFITS AND RISKS OF INCREASED SPATIAL RESOLUTION IN THE MANAGEMENT OF FISHERY METAPOPULATIONS UNDER UNCERTAINTY

Abstract Stock assessments and harvest guidelines are typically based on the concept of a “fish stock,” which may encompass a very large area. The presence of discrete subpopulations within managed fish stocks presents risks and opportunities for fishery management. Failure to manage catch at the same scale as the true population structure can lead to extirpation of discrete subpopulations and to declines in the productivity of the larger metapopulation. However, it may be difficult and costly to assess and manage stocks at a finer spatial scale, and there is likely greater uncertainty about the size of substocks than about the aggregate stock. We use a two‐area simulation model to compare the performance of fishery management at different spatial resolutions when there is uncertainty about growth, the size of the total population, and the relative size of the subpopulations. We show that relative benefits of finer scale management, in terms of profits and risks of depleting subpopulations, depend on a number of biological, technical, and economic factors. In some cases it may be both less risky and more profitable to manage the fishery with a single total allowable catch, even when there are biologically separate fish populations in the two areas.     

A JUVENILE‐ADULT DISCRETE‐TIME PRODUCTION MODEL OF EXPLOITED FISHERY SYSTEMS

Abstract. Previous mathematical modeling of the population dynamics of Georges Bank Atlantic cod fishery employed discrete‐time models without age‐structure. To make use of a much wider variety of data on fisheries and fish stocks than was possible with an unstructured model, we introduce a juvenile‐adult age‐structured production exploited fishery model with a very general recruitment function. We use the age‐structured model to study the interaction between fish exploitation levels and recruitment dynamics. As case studies, we use our model results and historical fish population data from Georges Bank to investigate the impact of recent harvesting levels on the sustainability of cod fishery. We show that a constant harvesting policy with the same harvesting rate of 2007 would lead to the recovery and sustainability of Georges Bank cod fishery.     

Catch‐to‐stock dependence: The case of small pelagic fishery with bounded harvesting effort

Biologic characteristics of schooling fish species explain why the rates of harvesting in pelagic fisheries are not proportional to the existent stock size and may exhibit no variation between the periods of fish abundance and scarcity. Therefore, the stock‐dependent nonlinearities in catchability must be reflected in the design of flexible fishing policies, which target the sustainable exploitation of this important natural resource. In this study, such nonlinearities are expressed through eventual variability of the “catch‐to‐stock” parameter that measures the sensitivity of an additional catch yield to marginal changes in the fish‐stock level. Using the optimal control modeling framework, we establish that each value of the “catch‐to‐stock” parameter generates a unique steady‐state size of the fish stock and the latter engenders an optimal fishing policy that can be sustained as long as the “catch‐to‐stock” parameter remains unchanged. We also prove the continuous dependence of the steady‐state stock and underlying fishing policy upon the mentioned “catch‐to‐stock” parameter and then focus on the analysis of the equilibrium responses to changes in this parameter induced by external perturbations. Recommendations for Resource Managers

• Marginal catches of pelagic fish stocks do not react in a linear way to changes in existing stock level, and the latter is captured in our model by the “catch‐to‐stock” parameter . Each observable value of engenders a unique steady‐state stock size that defines an optimal fishing policy, which can be sustained as long as remains unchanged.
• The ability of fishery managers to detect variations in the levels of hyperstability expressed by the “catch‐to‐stock” parameter may help them to anticipate new equilibrium responses in stock evolution and to make timely adjustments in the fishing policy.
• Plausible estimations of the “catch‐to‐stock” parameter , as well as detection of its possible alterations, can be carried out within the framework of Management Strategy Evaluation (MSE) approach where different data collected inside and outside the fishery are contrasted via the validation of a relatively simple decision‐making model (presented in this paper) coupled with other “operation models” of higher complexity.
• If the “catch‐to‐stock” parameter cannot be reasonably assessed (), the fishery managers may rely upon the lower bound of stationary stock size, which depends on economic and biological factors (such as the present and future economic values of the exploited fish stock, its marginal productivity, and underlying dynamics of biological growth).

     

Layoff costs and underutilization of labour in fisheries

The cost of reducing the labour force during a transition from an overexploited fishery to a bionomic fishery is taken into account. This affects both the long run steady state and the optimal approach to steady state. These effects are illustrated using the case of the NortheastArctic cod stock as a stylized example. The method outlined represents an operational way to assess harvest quotas as well as effort quotas both in the steady state and not least on the path to steady state. In the steady state analysis completely general functional forms are used, whereas in the optimal path analysis the objective function is required to be quadratic in the control variable. This requirement, however, incorporates the most important sources of nonlinearities such as downward sloping demand and increasing marginal costs.     

WELFARE MEASUREMENT AND COST-BENEFIT ANALYSIS IN NASH AND STACKELBERG DIFFERENTIAL FISH GAMES

ABSTRACT. This paper develops welfare measures and cost-benefit rules in Nash and Stackelberg differential fish games. The model consists of two fisheries competing on a common fishing ground, but the results are valid in more general settings. Under steady state conditions, the present value of the fisheries' future profit is directly proportional to the current profit plus the value of the net investments in the fish population (the Hamiltonian). Outside the steady state, the rule must, under a Nash game, be modified by adding the present value of the marginal harm done to the fishery by the competitor's future catches to the Hamiltonian. Under a Stackelberg game the present value of the leader's future profit remains proportional to the value of his Hamiltonian.     

THE VALUE OF SHORT‐RUN CLIMATE FORECASTS IN MANAGING THE COASTAL COHO SALMON (ONCORHYNCHUS KISUTCH) FISHERY IN WASHINGTON STATE

ABSTRACT. . In recent years our understanding of the intricate connections between climate variability, marine and freshwater environmental conditions and the responses of fish stocks has improved considerably. With predictable relationships between the environment and stock abundance, fishery managers should be able to forecast variation in stock survival and recruitment. Such forecasts present an opportunity for increasing the economic value of fisheries and for achieving other management objectives, such as stock conservation and maintenance of population diversity. After describing a 4‐step framework for addressing the question ‘What is a forecast worth?’ in a fishery decision‐making context, we introduce the management system for Washington's coastal coho salmon (Oncorhynchus kisutch) fishery. Then we apply the 4‐step framework to estimate the value of improved run size forecasts in the annual harvest management of coho salmon in Washington State. Our principal analytical tool is a stochastic simulation model that incorporates the main characteristics of the fishery. The paper concludes with a discussion of opportunities and constraints to the use of climate‐based forecasts in fishery management on various spatial and temporal scales, and we consider the challenges associated with forecasting variations in fish stock size caused by shifts in climate and related ocean conditions.     

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.     

HUMAN FERTILITY DECISIONS AND COMMON PROPERTY RESOURCES: A DYNAMIC ANALYSIS

ABSTRACT. In rural areas of developing countries, parental decisions on number of offspring may be made on the basis of the role of children in harvesting local common property renewable resources. It has been argued that this may lead to a cycle of human over‐population and resource over‐exploitation. To investigate the plausibility of this argument, we present a discrete dynamic model with two state variables representing human population level N and resource stock level S. The model is similar to one given by Nerlove and Meyer but differs in several important respects. It is assumed that, in each over‐lapping generation of parents and children, parents decide how many children to have based on their resulting share of the local resource harvest and the costs associated with child‐rearing. Using simulation and analytical methods, the long term steady state population and resource stock levels for this dynamic noncooperative game are contrasted with the steady state when parental fertility decisions are made in a cooperative manner.     

IN-SEASON FISHERY MANAGEMENT: A BAYESIAN MODEL

A Bayesian model is presented for optimizing harvest rates on an uncertain resource stock during the course of a fishing season. Pre-season stock status information, in the form of a “prior” probability distribution, is updated using new data obtained through the operation of the fishery, and harvest rates are chosen to achieve a balance between conservation concerns and fishing interests. A series of fishery scenarios are considered, determined by the stock size distribution and the timing distribution; the uncertainty in the fish stock is seen to have a rather complex influence on optimal harvest rates. The model is applied to a specific example, the Skeena River sockeye salmon fishery.     

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.     

OPTIMAL FISH HARVESTING FOR A POPULATION MODELED BY A NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATION

As the human population continues to grow, there is a need for better management of our natural resources in order for our planet to be able to produce enough to sustain us. One important resource we must consider is marine fish populations. We use the tool of optimal control to investigate harvesting strategies for maximizing yield of a fish population in a heterogeneous, finite domain. We determine whether these solutions include no‐take marine reserves as part of the optimal solution. The fishery stock is modeled using a nonlinear, parabolic partial differential equation with logistic growth, movement by diffusion and advection, and with Robin boundary conditions. The objective for the problem is to find the harvest rate that maximizes the discounted yield. Optimal harvesting strategies are found numerically.     

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.     

THE INCOMPLETE‐INFORMATION SPLIT‐STREAM FISH WAR: EXAMINING THE IMPLICATIONS OF COMPETING RISKS

ABSTRACT. . It is now widely recognized that climactic regime shifts, which aperiodically alter a harvested fish stock's biomass and spatial distribution, may lead to distorted fisheries management decisions which negatively impact the fishery, both biologically and economically. This is particularly true for trans‐boundary migratory stocks, where optimal management relies on coordination among independent nation‐states. Unanticipated changes in stock distribution and abundance can upset expectations of national authorities, leading them to sanction inappropriate harvesting levels by their separately managed fleets targeting the same breeding fish stock. Our theoretical studies are based on a spatially‐distributed stochastic model, which we have called the “split‐stream model,‘ where two separately managed fleets harvest simultaneously at two separate sites. Our key assumption is that competing fleet managers, when harvesting noncooperatively, hold incomplete and asymmetric private information of current stock recruitment and spatial distribution. When subsequently negotiating to coordinate their harvests, they agree that they will share their information and then bargain over partition of the gains from their cooperation. This bargaining process takes into account the fleet's relative competitive strengths, particularly due to private information asymmetries. In this present article we introduce a more complex information structure than had been assumed in our earlier work (McKelvey and Golubtsov [2002], McKelvey, Miller and Golubtsov [2003], Mckelvey et al. [2004]). Specifically, both stock‐growth and stock‐split parameters vary stochastically and asynchronously. Thus, when harvesting noncooperatively, each fleet may possess private knowledge which is unavailable to the other. We examine the interplay of the harvesting game's information structure with other fishery characteristics, such as the fleets' economics and operating characteristics and their attitudes toward risk, to determine the implications of such structure for the outcome of the harvesting game. All of these changes are made to capture new conceptual phenomena and expand the range of applicability of the model.     

Dynamics of a fishery system in a patchy environment with nonlinear harvesting

In this paper, a stock‐effort dynamical model with two fishing zones is discussed. The nonlinear harvesting function is assumed depending upon stock size as well as fishing effort. The migration of fish is considered between two zones. The harvesting vessels also move between zones to increase their revenue. The movements of fish and fishing vessels between zones are assumed to take place at a faster time scale as compared with processes involving growth and harvesting occurring at a slow time scale. The aggregated model is obtained for total fish stock and fishing effort. This aggregated (reduced) model is analyzed analytically as well as numerically. Biological and bionomic equilibria of the system are obtained, and criteria for local stability or instability of the system are derived. The impact of levels of taxation T on the fish population and on the revenue earned by the fishery is investigated. An optimal harvesting policy is also discussed using the Pontryagin's maximum principle. The aggregated model also exhibits Hopf and transcritical bifurcation with respect to the bifurcation parameter tax T. Numerical simulations are presented to illustrate the results.     

基于Leslie模型的资源持续利用

传统的Leslie模型是具有年龄结构种群演化的有效研究方法.基于持续收获状态的均匀收获是生物资源可持续利用的重要手段.对均匀收获状态下种群保持稳定的条件进行了研究,证明了均匀收获下稳定状态的充要条件,同时基于该条件,给出了临界稳定状态的判定方法,并就临界稳定状态下种群的分布情况进行了研究.通过引入实际数据并就实际数据进行计算,对该判定条件及最终的种群分布情况进行了实证分析.