HARVESTING STRATEGIES FOR A MIGRATORY FISH STOCK BASED ON A MODEL WITH UNCERTAINTY

We model a species of fish with breeding ground protected, and harvesting is carried out in an adjacent area. There is migration of fish from the breeding ground to the adjacent area. We demonstrate that in the presence of bounded uncertain disturbances, the strategy of harvesting the fish biomass at the maximum sustainable yield level is indeed optimal in the long term consideration.     

Global asymptotic stability of a two species competitive system with stage structure and harvesting

Introduction' There have recently appeared in the literature several mathematical models of stagestructured population growth, i. e., models which take into account the faCt that individuals in a population may belong to one of two classes, the immatures and the matureslllZI.Cannibalism has been observed in a great variety of species, including a number of fish species.Cannibalism models of various types have also been investigatedI3"l. In these models, the ageto maturity is represented by a…     

A study of harvesting in a predator–prey model with disease in both populations

Disease control by managers is a crucial response to emerging epidemics, and in the context of global change, emerging risks associated with parasites, invasive species, and infectious diseases are an important issue especially for developing countries. Our objective is to provide a mathematical framework to study the response of a predator–prey model to a disease in both populations and harvesting of prey species. We have worked out the conditions for local stability of the equilibrium points as well as persistence of the system. We have derived the ecological and disease basic reproduction numbers. These enable us to determine the community structure of the system. Harvesting may play a crucial role in a host–parasite system, and reasonable harvesting can remove parasite burden from the host. Our numerical results reveal that the reasonable harvesting prevents the oscillations of the species. We conclude that harvesting can be an effective strategy for controlling the spread of disease. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Positive Periodic Solution and Numerical Optimization in the Harvesting Effort for a Single-Species Stage-Structured System with Birth Pulses

In this paper, we consider a single-species stage-structured model with birth pulses and the harvesting of the species. Especially, we assume that the species can be divided into the immature and the mature, which exhibit different death rates. The mature species reproduces at fixed moments each year because the birth of many species is seasonal or occurs in a regular pulse, and the species is harvested not during the whole year but during a single period of the year. For such a system, we obtain conditions which guarantee the existence of a stable positive periodic solution. This implies that sustainable exploitation of the species can be maintained if we use the proper strategy in the harvesting effort and timing. Further, in order to get the maximum annual sustainable yield, we optimize the harvesting using numerical analysis; in addition, we find that the harvesting timing affects the maximum annual sustainable yield. Lastly, we show the effects of birth rate and harvesting effort on the dynamical complexity of the system with the help of a bifurcation graph.     

Stable coexistence mediated by specialist harvesting in a two zooplankton–phytoplankton system

This paper deals with a predator–prey model with specialist harvesting, representing a two predators (Zooplankton) and one resource (Phytoplankton) system. First, the existence and stability of equilibria is analyzed both from local and global point of view. Our results indicate that a specialist harvesting which is discriminate may mediate the coexistence of the two zooplankton species which competitively exclude each other in absence harvesting. Although in most cases increasing harvesting reduces the two zooplankton species numbers, when harvesting leads to coexistence, it may also lead to increase the two zooplankton species numbers. Furthermore, to protect fish population from over exploitation a control instrument tax is imposed. The problem of optimal taxation policy is then solved by using Pontryagin’s maximal principle. It is established that the zero discounting leads to the maximization of the net economic revenue to the society and an infinite discount rate leads to complete dissipation of the net economic revenue to the society. Finally, the impact of harvesting is mentioned along with numerical results to provide some support to the analytical findings.     

A prey-predator model with harvesting for fishery resource with reserve area

Considering that over exploitation would result in the extinction of the population, we propose and investigate a Holling II functional response prey-predator model with harvesting for fishery resource in a two-patch environment: a free fishing zone (patch 1) and a reserve zone (patch 2) where fishing is strictly prohibited. First, the presence of harvesting can impact the existence of equilibria. Further, stability criteria of the model is analyzed both from local and global point of view. Our results indicate that so long as the prey population in the reserved zone does not extinct, the both prey always exist, that is marine reserves should ensure the sustainability of system. Thus, marine reserves not only protect species inside the reserve area but they can also increase fish abundance in adjacent areas. Next, the existence of bionomic equilibrium and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle. It is established that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.     

The Stage-structured Predator-Prey System with Delay and Harvesting

In this article, we establish a mathematical model of two species with stage structure and the relation of predator-prey, obtain the necessary and sufficient condition for the permanence of two species and the extinction of one species or two species. We also obtain the optimal harvesting and the threshold of harvesting for the sustainable development and increasing delays can cause a bifurcation into periodic solutions. Finally, we give some numerical results and graphs.     

MULTISPECIES EXPLOITATION WITH EVOLUTIONARY SWITCHING OF HARVESTING STRATEGIES

In this paper, we propose a bioeconomic model which describes a fishery in which each of two noninteracting species is harvested by a given group of fishers during a defined time period. Then the Fishing Regulatory Authority allows each fisher to reconsider the harvesting decision at fixed (discrete) periods of time. The model derives from an Italian fisheries management experience in the Northern Adriatic Sea, where this kind of “self‐adjusting” fishing policy has been proposed to regulate harvesting of two shellfish species. The proposed dynamic model assumes the form of a hybrid system, as the natural growth functions of the two species (in continuous time) are coupled with a discrete time adaptive system that regulates how agents switch from one harvesting strategy to the other period by period according to an evolutionary mechanism based on profit comparison. In order to obtain some insights into the basic mechanisms of the system, some relevant benchmark cases are analyzed before tackling (mainly numerically) the complete hybrid model. Our results suggest that, for proper sets of parameters, this kind of myopic and adaptive self‐regulation may ensure a virtuous trade‐off between profit maximization and resource conservation, driven by cost externalities and market pressure.     

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 STAGE-STRUCTURED AND HARVESTING PREDATOR-PREY SYSTEM

A predator-prey system with independent harvesting in either species and BeddingtonDeAngelis functional response is investigated. By analyzing characteristic equations and using an iterative technique,we obtain a set of easily verifiable sufficient conditions,which ensure the local and global stability of the nonnegative equilibria of the system. It is also shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Numerical simulations are carried out t...     

On the stability and Hopf bifurcation of a prey-generalist predator system with independent age-selective harvesting

Age-selective harvesting where harvesting of species after a certain age is a scientific strategy with respect to biological and economical point of views. By this method we can overcome the unexpected extinction risk of any harvested population due to random harvesting below its maturation (age, body size or weight). The objective of this paper is to study dynamic behavior of preypredator system with alternative form of time delay in harvesting. Arino et al. [2] have given alternative expression for a delayed logistic equation. Using this expression of time delay, a preypredator system with Holling type III functional response and independent age-selective harvesting is proposed and analyzed. We find out the critical values of delay parameters under different dynamical situations and observe that system is stable and unstable when the delay parameters are bellow and above the critical values respectively and there is Hopf bifurcation when delay parameters cross the critical values. System shows these interesting dynamical features under different critical parametric restrictions. Using the normal form theory and the center manifold theorem, we determine the stability and direction of the bifurcating periodic solutions. Numerical simulations illustrate the analytical results.     

Food chain model with optimal harvesting in fuzzy environment

A multispecies harvesting model with mutual interactions is formulated based on Lotka–Voltera model with three competing species which are affected not only by harvesting but also by the presence of prey, predator and the third species, which is super predator. In order to understand the dynamics of the system, it is assumed that the super predator follows the logistic growth. Further, there is demand for all the above three species in the market and hence harvesting of all species is performed. We derive the condition for global stability of the system using a suitable Lyapunov function. The possibility of existence of bioeconomic equilibrium is discussed. The optimal harvest policy is studied and the solution is derived under imprecise inflation in fuzzy environment using Pontryagin’s maximal principle. Finally some numerical examples are discussed to illustrate the model.     

Bifurcations of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and prey harvesting

The dynamics of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and constant harvesting rate of prey are taken into account. The results developed in this article reveal far richer dynamics compared with the system without harvesting. We first make qualitative and bifurcation analysis of the system without harvesting and show that the system has a weak focus of multiplicity at most 2, at which a Hopf bifurcation occurs. However, the system with harvesting has four nonhyperbolic equilibria for some parameter values, such as two saddle-node, a cusp, and a weak focus of multiplicity at most 4, and exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension 2, and a Hopf bifurcation. It reveals that there exist some critical harvesting values such that the species are in danger of extinction when the harvesting rate is greater than the critical values, which indicates that the dynamics of the system are sensitive to the constant prey harvesting. Moreover, numerical simulations are presented to illustrate our theoretical results.     

Joint project of fishery and poultry - A bioeconomic model

This paper considers the optimal joint harvest of prawns and poultry in a linked bioeconomic system. Through the cultivation process, poultry and prawns are reciprocal predators of one another. Prawns of non-marketable quality are fed to the birds, and birds which perish (in greater numbers in the face of increased density) are fed to prawns, along with a lot of other things that one does not usually consider prawns to eat (hogs, broken rice, etc.). The paper derives optimality conditions for the joint “effort” imposed in each of these industries, where effort is somehow analogous to the control variable in classical Gordon-Schaefer fishery problems. Growth of both species is governed by parameters as well as externally applied nutrients and the biomass of the other species available as supplemental nutrition. Analysis of the boundedness of this dynamical system is discussed. The conditions for local and global stability are derived. Finally, an optimal harvesting policy is discussed by applying Pontryagin’s Maximal Principle. Due to linearity of the objective function with respect to the control variable, the solution is bang-bang in this control and the best policy is to reach the singular equilibrium as quickly as possible by switching to the singular control.     

互惠系统的最优脉冲捕获

对两种群互惠系统的脉冲捕获问题进行了全面的研究.确定了在被捕获种群达到一定数量时,再进行脉冲捕获的最优捕获策略.这样既维持了种群的生态平衡,又使我们在捕获过程中的获利最大.     

FROM HARVESTING TO NONHARVESTING UTILITY: AN OPTIMAL CONTROL APPROACH TO SPECIES CONSERVATION

The purpose of this paper is to retrace the evolution of mathematical models focused on relation and interaction between economic growth, sustainable development, and natural environment conservation. First, generic defensive expenditures are introduced into a common‐property harvesting model in order to favor the species growth. Second, a transition model comprising both harvesting and nonharvesting values of wildlife biological species emerges. The latter gives rise to a group of purely nonharvesting models where anthropic activities and economic growth may have positive or negative impact on the natural evolution of wildlife species. Several scholars have proved that optimal strategies that are relatively good for harvesting purposes are not simply “transferrable” to the context of conservation of wildlife biological species with no harvesting value. In addition, the existence of optimal policies for long‐term conservation of all biological species (with or without harvesting value) cannot be guaranteed without having relatively large species populations at the initial time. Therefore, all such strategies are incapable of enhancing the scarce populations of endangered species and, therefore, cannot save these species from eventual (local) extinction. As an alternative, policymakers may soon be compelled to design and implement short‐term defensive actions aimed at recovery and enhancement of endangered wildlife species.     

Optimal control of harvesting coupled with boundary control in a predator—prey system

We consider boundary control and control via harvesting in a parabolic predator—prey system for a bounded region. The boundary control depicts the relationship between the boundary environment and the possibly harmful species. In addition, a proportion of the predator is harvested for profit. We choose to maximize the objective functional which incorporates the amount of the prey and the revenue of harvesting of the predator less the economic cost of sustaining a satisfactory boundary habitat and the cost due to the harvesting component. Moreover, we characterize the unique optimal control in terms of the solution to the optimality system, which is the state system coupled with the adjoint system.     

Control theory and the management of ecosystems: A threshold policy with hysteresis is robust

Control theory is much used in engineering to stabilize a given dynamical system at a desired equilibrium point or to confine its trajectories to a region. Extinction of species is one of the most serious problems facing fishery and, to avoid it, different policies are applied. The threshold policy (TP) is a harvesting strategy commonly used in fisheries all over the world, and also is a special and simple case of the variable structure control (VSC). In this paper, the concept of virtual equilibrium point is used to design three different kinds of threshold policies; the standard one (TP), one with delay (TPD) and finally with hysteresis and delay (TPHD), for logistic models subject to Euler as well as two different nonstandard discretization schemes. Uncertainties in the intrinsic growth rate, in the carrying capacity, in the population density, and in the effort policy as well as an overexploitation situation are considered. A time lag in the control policies is also introduced. The important novel characteristic of the TPD and TPHD is that both ensure that, even though the system is subjected to uncertainties and a period of overexploitation, the system eventually stabilizes in bounded oscillations in a desired safe region of the state space. In addition, the sustainable yield under the proposed policies is compared with that of the proportional policy proposed in [1], in order to discuss economic aspects of the proposed threshold policies.     

DYNAMICS OF SPATIAL EXPLOITATION: A METAPOPULATION APPROACH

ABSTRACT. In this paper we present a bioeconomic model of a harvesting industry operating over a heterogeneous environment comprised of discrete biological populations interconnected by dispersal processes. The model generalizes the Gordon [1954]/Smith [1968] model of open-access rent dissipation by accounting for intertemporal and spatial “Ricardian” patterns of exploitation. This model yields a simple, but insightful, framework from which one can investigate factors that contribute to the evolution of resource exploitation patterns over space and time. For example, we find that exploitation patterns are driven by biological and fleet dispersal and biological and economic heterogeneity. We conclude that one cannot really understand the biological processes operating in an exploited system without knowing as much about the harvesting system as about the biological system.