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.     

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.     

Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process

In this paper, the stochastic harvesting problem is regarded as a mathematical formulation of finding the maximum sustained yield and the corresponding best sustainable harvesting strategies under uncertainty. We use a new method to solve this problem, and prove the equivalency between this method and previous methods. This paper is the first attempt to apply the ergodic theory on the optimal harvesting problem, to the best of our knowledge.     

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.     

模拟周期环境和尺度结构的种群系统的最优收获率

研究一类周期环境中基于个体尺度分布的种群资源模型的最优收获率问题.运用泛函分析方法证明了最优策略的存在性,借助非线性分析中的切锥-法锥技巧导出了最优策略的结构,为模型应用奠定了理论基础.     

The structure of optimal solutions for harvesting a renewable resource

We 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.

     

HARVESTING A TWO-PATCH PREDATOR-PREY METAPOPULATION

ABSTRACT. A mathematical model for a two-patch predator-prey metapoplation is developed as a generalization of single-species metapopulation harvesting theory. We find optimal harvesting strategies using dynamic programming and La-grange multipliers. If predator economic efficiency is relatively high, then we should protect a relative source prey subpopulation in two different ways: directly, with a higher escapement of the relative source prey subpopulation, and indirectly, with a lower escapement of the predator living in the same patch as the relative source prey subpopulation. Numerical examples show that if the growth of the predator is relatively low and there is no difference between prey and predator prices, then it may be optimal to harvest the predator to extinction. While, if the predator is more valuable compared to the prey, then it may be optimal to leave the relative exporter prey subpopulation unharvested. We also discuss how a ‘negative’ harvest might be optimal. A negative harvest might be considered a seeding strategy.     

Numerical Optimal Harvesting for a Periodic Age-Structured Population Dynamics with Logistic Term

We investigate an optimal harvesting problem for age-structured population dynamics with logistic term and periodic vital rates. We use first-order necessary optimality conditions in order to derive an algorithm to approximate the optimal harvesting effort. We present corresponding numerical experiments.     

Optimal harvesting policy of a stochastic predator–prey model with time delay

This paper concerns the optimal harvesting of a stochastic delay predator–prey model. Sufficient and necessary conditions for the existence of an optimal control are established. The optimal harvesting effort and the maximum value of the cost function are obtained as well. Some numerical tests are given to illustrate the main results.     

Quadratic harvesting dominated optimal strategy for a stochastic single-species model

A stochastic population model with the mixed harvesting strategy is formulated and studied in this paper. Sufficient and necessary conditions for survival of the species are derived firstly. Then, based on the ergodic stationary distribution, the optimal strategy is identified. Results show that the linear harvesting effort threatens to the survival of the species; the quadratic harvesting strategy occupies an absolute advantage in the harvesting and excludes the linear part out of the optimal harvesting strategy. It''s interest to see all these occur only in the random environments. Computer simulations are carried out to support the obtained results.     

Optimal harvesting policies for periodic Gompertz systems

In this paper, we study the periodic Gompertz system with harvesting. First, we analyze the system with continuous harvesting and obtain the maximum annual-sustainable yield, the optimal harvesting effort and the optimal population level for such a system. Then, the harvesting is assumed to occur at fixed moments every year, and we establish the Gompertz system with impulsive perturbation. And we investigate the impulsive harvesting policy to maximize the annual yield and to keep the population sustainable development. At last, the optimal results of the impulsive harvesting system are compared with those of the continuous harvesting system.     

OPTIMAL HARVESTING TIME OF FARMED AQUATIC POPULATIONS WITH NONLINEAR SIZE‐HETEROGENEOUS GROWTH

Abstract This paper examines the question of optimal harvesting time in a size‐heterogeneous farmed aquatic population, using a model reflecting the effect of population density on both overall mortality rate and individual growth. This analysis enables an optimal harvesting rule to be deduced. The results obtained are applied to shrimp culture in recirculation systems in Mexico. Numerical solutions are derived for different production scenarios. Assuming identical culture conditions, results are also obtained under the hypothesis of homogeneous population growth, the view traditionally taken in the relevant economic literature. The optimal harvesting times calculated tend to decrease with higher densities, although this rule fails under the size‐heterogeneous population model. In general, optimal harvesting times are overestimated when size‐homogeneity in the culture is assumed. Our analysis reveals that management predictions are significantly mistaken if the size‐heterogeneity phenomenon is not taken into account.     

一类半线性时变种群系统的最优捕获控制

讨论了年龄相关的半线性时变种群系统的最优捕获控制问题.根据微积分方程及泛函分析的知识证明了最优捕获控制的存在性,得到了捕获控制为最优的必要条件.     

The role of the water temperature in the optimal management of marine aquaculture

This paper analyses the influence of the water temperature over the optimal management of aquaculture farms. A fish growth model is presented and included in a general profit optimisation framework. Results reveal a positive influence of the average water temperature over the optimal harvesting time. The optimal ration size exhibits an uneven path with an upward phase in the warmer season followed by downward phase in winter time. An application for the seabream culture in the Mediterranean countries shows harvesting time and ration size very dependent on the water temperature and the stocking date. The risk involved with uncertainty of the environmental conditions is also estimated, which could explain the difference between the predicted optimal harvesting sizes and the real practice in commercial culture.     

Ordered Site Access and Optimal Forest Rotation

The ordered-site-access model of forest harvesting formulated for once-and-for-all forests in [7] is extended to the case of ongoing forests. The economic content of the corresponding optimal harvest schedule is delineated. For an infinite harvest sequence, the optimal schedule is shown to include the classical Faustmann rotation as a special case, and the effect of net revenue functions changing with harvest is studied. For the practically more important case of planning for a finite sequence of [INLINEEQUATION] harvests, the optimal harvest schedule is determined for a Faustmann environment with limited, and unlimited harvesting capacity, and its rapid convergence to the Faustmann rotation is shown for the case of unlimited harvesting capacity. The case of harvest cost functions varying with harvest rate is discussed. The existence of a steady-state optimal harvesting schedule (involving a pathwise uniform age distribution) for the more realistic Heaps-Neher environment and its relation to the Faustmann rotation are analyzed. The evolution of the optimal harvest schedule for a finite harvest sequence in a Heaps-Neher environment toward this steady-state (Faustmann type) rotation is demonstrated.     

Optimal Spatial Harvesting Strategy and Symmetry-Breaking

A reaction-diffusion model with logistic growth and constant effort harvesting is considered. By minimizing an intrinsic biological energy function, we obtain an optimal spatial harvesting strategy which will benefit the population the most. The symmetry properties of the optimal strategy are also discussed, and related symmetry preserving and symmetry breaking phenomena are shown with several typical examples of habitats.     

Optimal Harvesting for a Logistic Population Dynamics Driven by a Lévy Process

The optimal harvesting problem for a stochastic logistic jump-diffusion process is studied in this paper. Two kinds of environmental noises are considered in the model. One is called white noise which is described by a standard Brownian motion, and the other is called jumping noise which is described by a Lévy process. For three types of yield functions (time averaging yield, expected yield and sustainable yield), the optimal harvesting efforts, the corresponding maximum yields and the steady states of population mean under optimal harvesting strategy are respectively given. A new equivalent relationship among these three different objective functions is showed by the ergodic method. This method provides a new approach to the optimal harvesting problem. Results in this paper show that environmental noises have important effect on the optimal harvesting problem.     

食饵种群受毒素影响的三种群模型的最优收获问题

讨论了在毒素存在的情况下收获食饵的食饵—捕食模型的平衡点稳定性,生物经济平衡点的存在性和最优收获问题,利用Pontryagin极大值原理确定了最优收获策略.     

Optimal impulsive harvesting policy for single population

In this paper, we established the exploitation of impulsive harvesting single autonomous population model by Logistic equation. By some special methods, we analysis the impulsive harvesting population equation and obtain existence, the explicit expression and global attractiveness of impulsive periodic solutions for constant yield harvest and proportional harvest. Then, we choose the maximum sustainable yield as management objective, and investigate the optimal impulsive harvesting policies respectively. The optimal harvest effort that maximizes the sustainable yield per unit time, the corresponding optimal population levels are determined. At last, we point out that the continuous harvesting policy is superior to the impulsive harvesting policy, however, the latter is more beneficial in realistic operation.     

具有年龄结构和约束的群落系统的最优收获

研究一类有年龄结构和相互作用的两种群构成的群落系统的最优收获问题,要求控制过程结束时的种群状态充分接近预先指定的年龄分布.证明了最优控制的存在性,运用Dubovitskii-Milyutin理论导出了最优性条件.这种处理方法为研究连续年龄分布下种群收获问题提供了统一框架.