Influence of prey reserve in a prey–predator fishery

The excessive and unsustainable exploitation of marine resources has to led to the promotion of marine reserve as a fisheries management tool. In this paper we study a prey–predator system in a two-patch environment: one accessible to both prey and predators (patch 1) and the other one being a refuge for the prey (patch 2). The prey refuge (patch 2) constitutes a reserve zone of prey and fishing is not permitted, while the unreserved zone area is an open-access fishery zone. The existence of possible steady states, along with their local and global stability, is discussed. We then examine the possibilities of the existence of bionomic equilibrium. An optimal harvesting policy is given using Pontryagin’s maximum principle.     

Optimal harvesting of a Gompertz population model with a marine protected area and interval‐value biological parameters

To protect fishery populations on the verge of extinction and sustain the biodiversity of the marine ecosystem, marine protected areas (MPA) are established to provide a refuge for fishery resource. However, the influence of current harvesting policies on the MPA is still unclear, and precise information of the biological parameters has yet to be conducted. In this paper, we consider a bioeconomic Gompertz population model with interval‐value biological parameters in a 2‐patch environment: a free fishing zone (open‐access) and a protected zone (MPA) where fishing is strictly prohibited. First, the existence of the equilibrium is proved, and by virtue of Bendixson‐dulac Theorem, the global stability of the nontrivial steady state is obtained. Then, the optimal harvesting policy is established by using Pontryagin's maximum principle. Finally, the results are illustrated with the help of some numerical examples. Our results show that the current harvesting policy is advantageous to the protection efficiency of an MPA on local fish populations.     

FISHING YIELD,CURVATURE AND SPATIAL BEHAVIOR: IMPLICATIONS FOR MODELING MARINE RESERVES

ABSTRACT. Given a paucity of empirical data, policymakers are forced to rely on modeling to assess potential impacts of creating marine reserves to manage fisheries. Many modeling studies of reserves conclude that fishing yield will increase (or decrease only modestly) after creating a reserve in a heavily exploited fishery. However, much of the marine reserves modeling ignores the spatial heterogeneity of fishing behavior. Contrary to empirical findings in fisheries science and economics, most models assume explicitly or implicitly that fishing effort is distributed uniformly over space. This paper demonstrates that by ignoring this heterogeneity, yield‐per‐recruit models systematically overstate the yield gains (or understate the losses) from creating a reserve in a heavily exploited fishery. Conversely, at very low levels of exploitation, models that ignore heterogeneous fishing effort overstate the fishing yield losses from creating a reserve. Starting with a standard yield‐per‐recruit model, the paper derives a yield surface that maps spatially differentiated fishing effort into total long‐run fishing yield. It is the curvature of this surface that accounts for why the spatial distribution of fishing effort so greatly affects predicted changes from forming a reserve. The results apply generally to any model in which the long‐run fishing yield has similar curvature to a two‐patch Beverton‐Holt model. A simulation of marine reserve formation in the California red sea urchin fishery with Beverton‐Holt recruitment, eleven patches, and common larval pool dispersal dynamics reinforces these results.     

A MODEL OF TROPICAL MARINE RESERVE‐FISHERY LINKAGES

ABSTRACT. The excessive and unsustainable exploitation of our marine resources has led to the promotion of marine reserves as a fisheries management tool. Marine reserves, areas in which fishing is restricted or prohibited, can offer opportunities for the recovery of exploited stock and fishery enhancement. In this paper we examine the contribution of fully protected tropical marine reserves to fishery enhancement by modeling marine reserve‐fishery linkages. The consequences of reserve establishment on the long‐run equilibrium fish biomass and fishery catch levels are evaluated. In contrast to earlier models this study highlights the roles of both adult (and juvenile) fish migration and larval dispersal between the reserve and fishing grounds by employing a spawner‐recruit model. Uniform larval dispersal, uniform larval retention and complete larval retention combined with zero, moderate and high fish migration scenarios are analyzed in turn. The numerical simulations are based on Mombasa Marine National Park, Kenya, a fully protected coral reef marine reserve comprising approximately 30% of former fishing grounds. Simulation results suggest that the establishment of a fully protected marine reserve will always lead to an increase in total fish biomass. If the fishery is moderately to heavily exploited, total fishery catch will be greater with the reserve in all scenarios of fish and larval movement. If the fishery faces low levels of exploitation, catches can be optimized without a reserve but with controlled fishing effort. With high fish migration from the reserve, catches are optimized with the reserve. The optimal area of the marine reserve depends on the exploitation rate in the neighboring fishing grounds. For example, if exploitation is maintained at 40%, the ‘optimal’ reserve size would be 10%. If the rate increases to 50%, then the reserve needs to be 30% of the management area in order to maximize catches. However, even in lower exploitation fisheries (below 40%), a small reserve (up to 20%) provides significantly higher gains in fish biomass than losses in catch. Marine reserves are a valuable fisheries management tool. To achieve maximum fishery benefits they should be complemented by fishing effort controls.     

基于食饵保护的食饵-捕食捕鱼模型分析(英文)

In this paper,we consider a prey-predator fishery model with prey dispersal in a two-patch environment,one is assumed to be a free fishing zone and the other is a reserved zone where fishing and other extractive activities are prohibited.The existence of possible steady states of the system is discussed.The local and global stability analysis has been carried out.An optimal harvesting policy is given using Pontryagin s maximum principle.     

ECONOMICALLY OPTIMAL MARINE RESERVES WITHOUT SPATIAL HETEROGENEITY IN A SIMPLE TWO‐PATCH MODEL

Bioeconomic analyses of spatial fishery models have established that marine reserves can be economically optimal (i.e., maximize sustainable profit) when there is some type of spatial heterogeneity in the system. Analyses of spatially continuous models and models with more than two discrete patches have also demonstrated that marine reserves can be economically optimal even when the system is spatially homogeneous. In this note we analyze a spatially homogeneous two‐patch model and show that marine reserves can be economically optimal in this case as well. The model we study includes the possibility that fishing can damage habitat. In this model, marine reserves are necessary to maximize sustainable profit when dispersal between the patches is sufficiently high and habitat is especially vulnerable to damage.     

A two-patch model for the optimal management of a fishing resource considering a marine protected area

In this work, we propose and analyze a model related with the management optimization of a renewable resource in aquatic environment composed of two different patches. Spatial distribution of each subpopulation is assumed: one is developed in a marine protected area (MPA) or a marine reserve and the other is located in a zone where fishing with open access may be effected.It is generally assumed that there may be migration between both areas, but in this work we will consider that the flux goes.When a fishing ban in the protected area is established it becomes a marine reserve, which can also be assumed as a refuge for the captured species. In this case, the marine reserve is the source and the exploitation area is a sink.The behavior of the renewable resource is modeled by a deterministic continuous time system. To establish the optimal harvesting policy, we will maximize the present value J of a continuous time stream of revenues, given by a cost functional indicating the net economic revenue to the fishermen, the perceived rent. Using Pontragyn’s Maximum Principle we will obtain the Hamiltonian function to determine the optimal policies.     

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.     

THE ECONOMICS OF MARINE RESERVES

ABSTRACT. Marine reserves can be a useful supplement to other methods of fisheries management, but marine reserves alone are not likely to achieve a great deal in economic terms andperhaps not even in terms of conservation. The effects of marine reserves with open access elsewhere are analyzed, using a logistic model for a population with a patchy distribution. It is assumedthat a marine reserve is establishedfor the territory of one of two sub‐populations which interact through migrations. The total population increases while the total catch declines for the most part. A high rate of migration would, however, dilute the conservation effect. Examining a stochastic variant of the model shows that the variability (sum of squareddeviations) of catches may decrease as a result of protecting one of the sub‐populations. Even if all rents disappear by assumption, it is possible to identify this as an economic benefit, particularly when the average catch increases. The variability of the catch falls for a range of values of the population migration parameter and variability of growth, both when the stochastic disturbances are independent and when they are perfectly correlated for the two sub‐populations, andalso when the growth variability parameter differs between the sub‐populations.     

有保护区的海洋渔业资源离散动力学模型分析

假设海洋渔业资源分属于保护区和非保护区两个区域,本文建立一个渔业资源储量-捕捞力度模型,用聚合方法得到一个简化的离散动力系统,从而分析正不动点的存在性、稳定性以及关于保护区面积比例的局部分叉,运用中心流形定理分析平衡点的局部稳定性,并用数值模拟验证不动点的局部分叉.最后,用全局分析方法分析保护区面积比例变化对可行吸引域的结构和大小的影响,从而揭示保护区对渔业资源可持续利用的影响.     

CREATION OF MARINE RESERVES AND INCENTIVES FOR BIODIVERSITY CONSERVATION

Abstract Despite a number of benefits, marine reserves provide neither incentives for fishermen to protect biodiversity nor compensation for financial loss due to the designation of the reserves. To obtain fishermen's support for marine reserves, some politicians have suggested that managers of new marine reserves should consider subsidizing or compensating those fishermen affected by the new operations. The objective of this paper is to apply principal–agent theory, which is still infrequently applied to fisheries, to define the optimal reserve area, fishing effort, and transfer payments in the context of symmetric and asymmetric information between managers and fishermen. The expected optimal reserve size under asymmetric information is smaller than that under symmetric information. Fishing efforts encouraged with a transfer payment are always less compared to those without payment. This reflects the fact that as the manager induces the fishermen to participate in the conservation program, the fishermen will take into account their effects on fish stock by decreasing their effort. Examples are also supplied to demonstrate these concepts.     

Optimal Harvesting in an Age-Structured Predator–Prey Model

We investigate optimal harvesting control in a predator–prey model in which the prey population is represented by a first-order partial differential equation with age-structure and the predator population is represented by an ordinary differential equation in time. The controls are the proportions of the populations to be harvested, and the objective functional represents the profit from harvesting. The existence and uniqueness of the optimal control pair are established.     

Bifurcation and control of a bioeconomic model of a prey–predator system with a time delay

In this paper, we analyze the dynamical behaviour of a bioeconomic model system using differential algebraic equations. The system describes a prey–predator fishery with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is a protected zone. It is observed that a singularity-induced bifurcation phenomenon appears when a variation of the economic interest of harvesting is taken into account. We have incorporated a state feedback controller to stabilize the model system in the case of positive economic interest. A discrete-type gestational delay of predators is incorporated, and its effect on the dynamical behaviour of the model is analyzed. The occurrence of Hopf bifurcation of the proposed model with positive economic profit is shown in the neighbourhood of the coexisting equilibrium point through considering the delay as a bifurcation parameter. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.     

CONSERVATION OF WILDLIFE. A BIO‐ECONOMIC MODEL OF A WILDLIFE RESERVE UNDER THE PRESSURE OF HABITAT DESTRUCTION AND HARVESTING OUTSIDE THE RESERVE

ABSTRACT. Biodiversity is today threatened by many factors of which destruction and reduction of habitats are considered most important for terrestrial species. One way to counteract these threats is to establish reserves with restrictions on land use and exploitation. However, very few reserves can be considered islands, wildlife species roam over large expanses, often via some density dependent dispersal process. As a consequence, habitat destruction, and exploitation, taking place outside will influence the species abundance inside the conservation area. The paper presents a theoretical model for analyzing this type of management problem. The model presented allows for both the common symmetric dispersal as well as what is called asymmetric dispersal between reserve and outside area. The main finding is that habitat destruction outside may not necessarily have negative impact upon the species abundance in the reserve. As a consequence, economic forces working in the direction of reducing the surrounding habitat have unclear effects on the species abundance within the protected area. We also find that harvesting outside the reserve may have quite modest effect on the species abundance in the reserve. This underlines the attractiveness of reserves from a conservation viewpoint.     

A diffusive predator-prey model with a protection zone

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue of the Laplacian operator over Ω₀ with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator-prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.     

Homoclinic bifurcation for a general state-dependent Kolmogorov type predator–prey model with harvesting

In this paper, a general Kolmogorov type predator–prey model is considered. Together with a constant-yield predator harvesting, the state dependent feedback control strategies which take into account the impulsive harvesting on predators as well as the impulsive stocking on the prey are incorporated in the process of population interactions. We firstly study the existence of an order-1 homoclinic cycle for the system. It is shown that an order-1 positive periodic solution bifurcates from the order-1 homoclinic cycle through a homoclinic bifurcation as the impulsive predator harvesting rate crosses some critical value. The uniqueness and stability of the order-1 positive periodic solution are derived by applying the geometry theory of differential equations and the method of successor function. Finally, some numerical examples are provided to illustrate the main results. These results indicate that careful management of resources and harvesting policies is required in the applied conservation and renewable resource contexts.     

Novel dynamics of a predator–prey system with harvesting of the predator guided by its population

A predator–prey model was extended to include nonlinear harvesting of the predator guided by its population, such that harvesting is only implemented if the predator population exceeds an economic threshold. The proposed model is a nonsmooth dynamic system with switches between the original predator-prey model (free subsystem) and a model with nonlinear harvesting (harvesting subsystem). We initially examine the dynamics of both the free and the harvesting subsystems, and then we investigate the dynamics of the switching system using theories of nonsmooth systems. Theoretical results showed that the harvesting subsystem undergoes multiple bifurcations, including saddle-node, supercritical Hopf, Bogdanov–Takens and homoclinic bifurcations. The switching system not only retains all of the complex dynamics of the harvesting system but also exhibits much richer dynamics such as a sliding equilibrium, sliding cycle, boundary node (saddle point) bifurcation, boundary saddle-node bifurcation and buckling bifurcation. Both theoretical and numerical results showed that, by implementing predator population guided harvesting, the predator and prey population could coexist in more scenarios than those in which the predator may go extinct for the continuous harvesting regime. They could either stabilize at an equilibrium or oscillate periodically depending on the value of the economic threshold and the initial value of the system.     

On non-selective harvesting of a multispecies fishery

The present paper deals with the problem of non-selective harvesting of a prey-predator system in which both the prey and the predator species obey the law of logistic growth and each predators functional response to the prey approaches a constant as the prey population increases. Boundedness of the exploited system is examined. The existence of its steady states and their stability are studied using eigenvalue analysis. The existence of bionomic equilibria has been considered. The problem of determining the optimal harvest policy is then solved by using Pontryagin's maximal principle. Finally, some numerical examples are given to illustrate the results.     

THE IMPACTS OF MARINE RESERVES ON LIMITED‐ENTRY FISHERIES

ABSTRACT. We utilize a spatial bioeconomic model to investigate the impacts of creating reserves on limited‐entry fisheries. We find that reserve creation can produce win‐win situations where aggregate biomass and the common license (lease) price increase. These situations arise in biological systems where dispersal processes are prevalent and the fishery prior to reserve creation is operating at effort levels in a neighborhood of open‐access levels. We also illustrate that using strictly biological criteria for siting reserves (e.g., setting aside the most biological productive areas) will likely induce the most vociferous objections from the fishing industry. In general, we find that the dispersal rate and the degree the patches are connected play a significant role on the net impacts on the fishing sector.