Optimal Harvesting Problem Research Articles

The switch point algorithm applied to a harvesting problem.

In this paper, we investigate an optimal harvesting problem of a spatially explicit fishery model that was previously analyzed. On the surface, this problem looks innocent, but if parameters are set to where a singular arc occurs, two complex questions arise. The first question pertains to Fuller's phenomenon (or chattering), a phenomenon in which the optimal control possesses a singular arc that cannot be concatenated with the bang-bang arcs without prompting infinite oscillations over a finite region. 1) How do we numerically assess whether or not a problem chatters in cases when we cannot analytically prove such a phenomenon? The second question focuses on implementation of an optimal control. 2) When an optimal control has regions that are difficult to implement, how can we find alternative strategies that are both suboptimal and realistic to use? Although the former question does not apply to all optimal harvesting problems, most fishery managers should be concerned about the latter. Interestingly, for this specific problem, our techniques for answering the first question results in an answer to the the second. Our methods involve using an extended version of the switch point algorithm (SPA), which handles control problems having initial and terminal conditions on the states. In our numerical experiments, we obtain strong empirical evidence that the harvesting problem chatters, and we find three alternative harvesting strategies with fewer switches that are realistic to implement and near optimal.

Stochastic Volterra equations with time-changed Lévy noise and maximum principles

Motivated by a problem of optimal harvesting of natural resources, we study a control problem for Volterra type dynamics driven by time-changed Lévy noises, which are in general not Markovian. To exploit the nature of the noise, we make use of different kind of information flows within a maximum principle approach. For this we work with backward stochastic differential equations (BSDE) with time-change and exploit the non-anticipating stochastic derivative introduced in Di Nunno and Eide (Stoch Anal Appl 28:54-85, 2009). We prove both a sufficient and necessary stochastic maximum principle.

Dynamics and optimal harvesting of prey–predator in a polluted environment in the presence of scavenger and pollution control

This paper is concerned with the dynamics and optimal harvesting of a prey–predator system in a polluted environment in the presence of scavengers and pollution control. Toxicants, released from external sources and the dead bodies of prey and predators, pollute the environment, which affects the growth of both prey and predators, resulting in a decline in the economic revenue from harvest. We assume that scavengers reduce pollution by consuming dead bodies. Further, we consider pollution reduction through depollution efforts as an alternative to enhancing revenue. We propose and analyze a prey–predator–pollutant model and study the optimal harvesting problem. We investigate the persistence of the ecosystem, and we solve the optimal harvest problem using Pontryagin’s maximum principle. The results indicate that uncontrolled prey harvesting and a high rate of pollution drive the system toward the extinction of both species. A moderate amount of pollution and the reasonable harvest efforts allow the system to persist. The optimal harvest strategy highlights that investing in pollution reduction enhances the persistence of the system as well as economic revenue. Numerical examples demonstrate the significant outcomes of the study.

Exploration on dynamics in a ratio-dependent predator-prey bioeconomic model with time delay and additional food supply.

In this manuscript, a novel ratio-dependent predator-prey bioeconomic model with time delay and additional food supply is investigated. We first change the bioeconomic model into a normal version by virtue of the differential-algebraic system theory. The local steady-state of equilibria and Hopf bifurcation could be derived by varying time delay. Later, the formulas of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions are obtained by the normal form theory and the center manifold theorem. Moreover, employing the Pontryagin's maximum principle and considering the instantaneous annual discount rate, the optimal harvesting problem of the model without time delay is analyzed. Finally, four numeric examples are carried out to verify the rationality of our analytical findings. Our analytical results show that Hopf bifurcation occurs in this model when the value of bifurcation parameter, the time delay of the maturation time of prey, crosses a critical value.

The structure of optimal solutions for harvesting a renewable resource

AbstractWe consider the problem of optimal harvesting of a renewable resource whose dynamics are governed by logistic growth and whose payoff is proportional to the harvest. We consider both the case of a finite and an infinite time horizon and analyse the structure of the optimal solutions and their dependence on the parameters of the model. We show that the optimal policy can only have one of three structures: (1) maximal harvesting effort until the resource is depleted, (2) zero harvesting during an initial time interval followed by a subsequent switch to maximal harvesting effort, or (3) a singular solution, which corresponds to an intermediate level of harvesting, accompanied by the most rapid approach path. All three scenarios emerge, with minor variations, with finite and infinite time horizons, depending on the particular combination of parameters of the system. We characterize the conditions under which the singular solution is optimal and present suggestions for designing an optimal and sustainable harvesting strategy.Recommendations for Resource Managers: We have rigorously explored a standard optimal harvesting model and its steady states. We show that three different types of solutions may emerge: (i) maximal harvesting eventually leading to a complete depletion of the stock; (ii) maximal harvesting with a potential period of idleness leading to a positive stock; (iii) an initial phase of either no or full harvesting followed by a period of intermediate harvesting intensity leading to a positive stock (singular solution). With some modifications, similar results hold for a finite planning horizon. Which of these three scenarios emerges in the finite horizon case depends not only on the parameter values but also on the length of the planning horizon.

Optimal harvesting for a predator‐prey system with age structure and reserve area

Considering that the creation of marine reserves is one potential solution to the over exploitation of marine life and different population age implies different reproduction as well as survival capacities, we propose an age‐structured predator‐prey model for fishery resource with reserve area where fishing is strictly prohibited, and then consider the corresponding optimal harvesting problem where the objective functional represents the total economic benefits obtained from the harvesting process. First, the existence and uniqueness of the solution of the model are proved by means of the fixed point theorem. Then, the optimality conditions for achieving the optimal harvesting policy are given by solving the extremum point of the cost functional. Furthermore, the existence of a unique optimal harvesting strategy is demonstrated with the use of Ekeland's Principle. Finally, the numerical simulations indicate that the creation of marine reserves is indeed conducive to the recovery of fishery resources, increasing fish abundance and protecting biodiversity as well as ecosystem structure.

Optimal harvesting in a unidirectional consumer–resource mutualisms system with size structure in the consumer

This paper considers the optimal harvesting problem for a size-structured model of unidirectional consumer–resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer. First, we show the existence of a unique nonnegative solution of the system and give the continuous dependence of solutions on the control variable. Next, the adjoint system is derived, which is necessary for optimality and the existence of a unique optimal policy. Then necessary conditions for optimality are established via the normal cone and adjoint system. Moreover, the existence of a unique optimal strategy is proved via Ekeland’s variational principle and fixed-point reasoning in convex analysis. Finally, we use numerical simulations to verify the main results and find other dynamic properties of the system.

Maximum principle for the optimal harvesting problem of a size-stage-structured population model

<p style='text-indent:20px;'>The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.</p>

Optimal harvesting for a periodic competing system with size structure in a polluted environment

<abstract><p>As a renewable resource, biological population not only has direct economic value to people's lives, but also has important ecological and environmental value. This study examines an optimal harvesting problem for a periodic, competing hybrid system of three species that is dependent on size structure in a polluted environment. The existence and uniqueness of the nonnegative solution are proved via an operator theory and fixed point theorem. The necessary optimality conditions are derived by constructing an adjoint system and using the tangent-normal cone technique. The existence of unique optimal control pair is verified by means of the Ekeland variational principle and a feedback form of the optimal policy is presented. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. The objective functional represents the total profit obtained from harvesting three species. The results obtained in this work can be extended to a wide variety of fields.</p></abstract>

Optimal harvesting for a periodic n-dimensional food chain model with size structure in a polluted environment.

This study examines an optimal harvesting problem for a periodic n-dimensional food chain model that is dependent on size structure in a polluted environment. This is closely related to the protection of biodiversity, as well as the development and utilization of renewable resources. The model contains state variables representing the density of the ith population, the concentration of toxicants in the ith population, and the concentration of toxicants in the environment. The well-posedness of the hybrid system is proved by using the fixed point theorem. The necessary optimality conditions are derived by using the tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal control pair are verified via the Ekeland variational principle. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. Some numerical tests are given to illustrate that the numerical solution has good periodicity. The objective functional here represents the total profit obtained from harvesting n species.

Stability and bifurcation analysis of discrete predator–prey model with nonlinear prey harvesting and prey refuge

In this paper, a discrete Leslie–Gower model with nonlinear prey harvesting and prey refuge is proposed. We prove the existence and stability of an interior equilibrium of the model. In addition, the existence of flip bifurcation and Neimark–Sacker bifurcation at the interior equilibrium is studied by applying central manifold theory and bifurcation theory. Using Pontryagin's maximum principle, we propose an optimal harvesting problem.

Reinforcement learning in optimizing forest management

We solve a stochastic high-dimensional optimal harvesting problem by using reinforcement learning algorithms developed for agents who learn an optimal policy in a sequential decision process through repeated experience. This approach produces optimal solutions without discretization of state and control variables. Our stand-level model includes mixed species, tree size structure, optimal harvest timing, choice between rotation and continuous cover forestry, stochasticity in stand growth, and stochasticity in the occurrence of natural disasters. The optimal solution or policy maps the system state to the set of actions, i.e., clear-cutting, thinning, or no harvest decisions as well as the intensity of thinning over tree species and size classes. The algorithm repeats the solutions for deterministic problems computed earlier with time-consuming methods. Optimal policy describes harvesting choices from any initial state and reveals how the initial thinning versus clear-cutting choice depends on the economic and ecological factors. Stochasticity in stand growth increases the diversity of species composition. Despite the high variability in natural regeneration, the optimal policy closely satisfies the certainty equivalence principle. The effect of natural disasters is similar to an increase in the interest rate, but in contrast to earlier results, this tends to change the management regime from rotation forestry to continuous cover management.

Optimal control for a size-structured predator-prey model in a polluted environment

In this paper, we deal with an optimal harvesting problem for a periodic predator-prey hybrid system dependent on size-structure in a polluted environment. In other words, a size-dependent model in an environment with a small toxicant content has been established. The well-posedness of state system is proved by using the fixed point theorem. The necessary optimality conditions are derived by tangent-normal cone technique in nonlinear functional analysis. The existence of a unique optimal harvesting policy is verified via the Ekeland’s variational principle. The optimal harvesting problem has an optimal harvesting policy, which has a Bang-Bang structure and provides a threshold for the optimal harvesting problem. Using the optimization theories and methods in mathematics to control phenomena of life. The objective function represents the total economic profit from the harvested population. Some theoretical results obtained in this paper provide a scientific theoretical basis for the practical application of the model.

Measure valued solutions for an optimal harvesting problem

The paper deals with the harvesting model introduced by one of the authors in a joint paper with A. Bressan and W. Shen. A cost term which plays an essential role in the existence proof is removed here leading to some new phenomena. Since a solution not always exists in this case, we propose a relaxed version of the problem which is always solvable, has a geometrical interpretation in the functions space and whose unique solution, under suitable conditions, solves also the original problem. Finally, the notion of strategy limit of a sequence of smooth measures is introduced as a long-term control to fish populations.

A symplectic pseudospectral method for constrained time-delayed optimal control problems and its application to biological control problems

ABSTRACT Time-delays are often involved in modelling biological processes, which arise more difficulties for the control task. In fact, once the control goal is given, a constrained optimal control problem can be formulated to describe the biological control mission. However, it’s almost impossible to obtain analytical solutions for such complicated problems, especially for those with high nonlinearity and complex constraints. In this paper, a symplectic pseudospectral method for solving nonlinear optimal control problems of constrained stated-delayed system is developed. The original nonlinear problem is transformed into a series of linear-quadratic optimal control problems by the successive convexification technique at the first implementation. By applying the local Legendre-Gauss-Lobatto pseudospectral method and the parametric variational principle, each of the transformed linear-quadratic problems is converted into a standard linear complementarity problem which can be solved easily by the Lemke’s method. The cost functional, system dynamics and constraints can be generally nonlinear functions of state, control and time variables. and three kinds of inequality constraints, i.e. pure-state, pure-control and mixed state-control, can be treated under a uniform framework. The developed symplectic pseudospectral method is finally validated by an optimal harvesting problem and an optimal vaccination problem.

Analyticity of free boundary in an optimal harvesting problem

This note revisits an optimisation problem pertaining to the optimal harvesting of a marine species. The existence of solutions and the corresponding optimal conditions they satisfy have already been proved. It is known that the optimal solutions can be identified with n-dimensional shapes. We will obtain an interesting result concerning the free boundary of the optimal shapes. Indeed, we will prove that if a parameter in the admissible set is kept sufficiently small then the free boundaries will be real analytic hypersurfaces.

Global analysis and optimal harvesting for a hybrid stochastic phytoplankton-zooplankton-fish model with distributed delays.

In this paper, we formulate a phytoplankton-zooplankton-fish model with distributed delays and hybrid stochastic noises involving Brownian motion and Markov chain, and propose an optimal harvesting problem pursuing the maximum of total economic income. By global analysis in terms of some system parameters, we investigate the dynamical behaviors on the well-posedness, bounded- ness, persistence, extinction, stability and attractiveness of the solutions for the stochastic delayed system. Moreover, we provide sufficient and necessary condition ensuring the existence of the optimization solution for the optimization problem and obtain the optimal harvesting effect and the maximum of sustainable yield. Lastly, two numerical examples and their simulations are given to illustrate the effectiveness of our results.

Optimal Harvesting for Size-Dependent Control

We investigate some optimal harvesting problems concerning the age-structured population dynamics. The goal is to maximize the profit with respect to a size-dependent harvesting effort. The harvesting is made either by fishing or by hunting. For each studied problem, we state the necessary optimality conditions and use them to get the structure of the optimal control. For the problem related to hunting, we derive an iterative algorithm to improve at each step the harvesting effort, in order to increase the profit. Numerical tests are performed to show that the proposed methods are effective.

Optimal harvesting of a hierarchical age-structured population system

We investigate an optimal harvesting problem for age-structured species, in which elder individuals are more competitive than younger ones, and the population is modeled by a highly nonlinear integro-partial differential equation with a global feedback boundary condition. The existence of optimal strategies is established by means of compactness and maximizing sequences, and the maximum principle obtained via an adjoint system, tangent-normal cones and a new continuity result. In addition, some numerical experiments are presented to show the effects of the price function and younger’s weight on the optimal profits.

Optimal harvesting of prey-predator fishery modeling in a two patch environment and harvesting in unprotected area

This article deals with the dynamics of prey and predator populations in a two patch environment, a protected area and an unprotected area for fishing. The prey disperses between the two patches and migrates easily. There are two predators, one is in the protected area and another is in the unprotected area. The predators cannot migrate. Both prey and predator in unprotected area are harvested with constant efforts. The dynamical behavior of the populations is stated as a system of differential equations. The existence of a positive equilibrium point and its stability are investigated. We discuss the local stability of the positive equilibrium point. The stable equilibrium point is then associated with optimal harvesting problems. Based on the analysis, we found that there exist a stable positive equilibrium point when there is no harvesting. For model with constant efforts for both prey and predator, we found that over fishing will maximize the profit but the predator in the unprotected area will be extinct. With the help of Pontryagin’s maximum principle in maximizing the present value of revenues, we found the extremal of the efforts that maximize the present value of revenues. This means that both prey and predator in the protected area as well as the prey and predator in the unprotected area are possibly coexist although the prey and the predator in the unprotected area are harvested with constant efforts. Some numerical simulations area given to confirm the result of analysis.