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.     

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.     

FORECASTING TREE SIZE DISTRIBUTION BY NUMERICAL APPROXIMATION OF THE FORWARD KOLMOGOROV EQUATION

ABSTRACT. Forest management planning of uneven‐aged stands involves forecasting of the tree size distribution. The temporal development of the size distribution in a forest stand may be described by the forward Kolmogorov equation. The objective of this study is to illustrate that numerical approximation of the solution to the equation provides a reasonably accurate way of forecasting future tree size distribution, especially for stands with non‐normal size distribution. Furthermore, a method for the practical application is devised. The analyses compare observed and forecasted tree size distributions for two forest stands, 1) an unthinned stand of Sitka spruce (Picea sitchensis (Bong.) Carr.), and 2) an uneven‐aged stand of beech (Fagus sylvatica L.) managed under the selection system. The analyses show that the size distribution in the uneven‐aged stand may be forecasted correctly for a 20 25 year period, while for the even‐aged stand the method seems to fail after 10 to 15 years.     

A fourth‐order method for numerical integration of age‐ and size‐structured population models

In many applications of age‐ and size‐structured population models, there is an interest in obtaining good approximations of total population numbers rather than of their densities. Therefore, it is reasonable in such cases to solve numerically not the PDE model equations themselves, but rather their integral equivalents. For this purpose quadrature formulae are used in place of the integrals. Because quadratures can be designed with any order of accuracy, one can obtain numerical approximations of the solutions with very fast convergence. In this article, we present a general framework and a specific example of a fourth‐order method based on composite Newton‐Cotes quadratures for a size‐structured population model. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

OPTIMAL GRAZING MANAGEMENT RULES IN SEMI‐ARID RANGELANDS WITH UNCERTAIN RAINFALL

Abstract. We study optimal adaptive grazing management under uncertain rainfall in a discrete‐time model. As in each year actual rainfall can be observed during the short rainy season, and grazing management can be adapted accordingly for the growing season; the closed‐loop solution of the stochastic optimal control problem does not only depend on the state variable, but also on the realization of the random rainfall. This distinguishes optimal grazing management from the optimal use of most other natural resources under uncertainty, where the closed‐loop solution of the stochastic optimal control problem depends only on the state variables. Solving this unusual stochastic optimization problem allows us to critically contribute to a long‐standing controversy over how to optimally manage semi‐arid rangelands by simple rules of thumb.     

Optimal harvest strategy for even‐aged stands with price uncertainty and risk of natural disturbances

We present a reservation price model to examine the joint impacts of natural disturbances and stumpage price uncertainty on the optimal harvesting decision for even‐aged forest stands. We consider a landowner who manages a loblolly pine stand to produce timber and amenities, under age‐dependent risk of wildfires and uncertainty in future timber prices. We show that the incorporation of risk of wildfires decreases the optimal reservation prices. The inclusion of risk of wildfires leads to lower land values and reduces the mean harvest age compared with the case of no‐risk of wildfires. Higher economic gains are obtained with the reservation price strategy compared with the deterministic rotation age model—a difference in the land value of $2,326 ha^?1 (21%) between the two approaches. Recommendations for Resource Managers

• Our adaptive harvest strategy shows that the incorporation of risk of wildfires decreases the optimal reservation prices compared with the case of no‐risk of wildfires.
• Low reservation prices—a price that makes the landowner indifferent between harvesting or waiting longer—result in lower economic benefits for landowners and potential conversions of lands to nonforest use.
• Forest management practices oriented to reduce the effects of catastrophic disturbances, for example, creating a more complex forest structure with different stand densities, become imperative to ensure the sustainability of forestlands in the US South.
• Our analysis also suggests that the valuation of forestry investments should consider not only the risk of catastrophic events but also uncertainty in future timber prices. Higher appraisals of land value are obtained when timber price uncertainty is explicitly recognized, providing financial incentives for landowners to invest in forestry.

     

Subspace‐by‐subspace preconditioners for structured linear systems

We consider the iterative solution of symmetric positive‐definite linear systems whose coefficient matrix may be expressed as the outer product of low‐rank terms. We derive suitable preconditioners for such systems, and demonstrate their effectiveness on a number of test examples. We also consider combining these methods with existing techniques to cope with the commonly‐occuring case where the coefficient matrix is the linear sum of elements, some of which are of very low rank. Copyright © 1999 John Wiley & Sons, Ltd.     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.     

A robust algebraic multilevel preconditioner for non‐symmetric M‐matrices

Stable finite difference approximations of convection‐diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M‐matrix, which is highly non‐symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non‐symmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis that applies to constant coefficient problems with periodic boundary conditions whenever using an ‘idealized’ version of the two‐level preconditioner. Within this setting, it is proved that any eigenvalue λ of the preconditioned system satisfies for some real constant c such that . This result holds independently of the grid size and uniformly with respect to the ratio between convection and diffusion. Extensive numerical experiments are conducted to assess the convergence of practical two‐ and multi‐level schemes. These experiments, which include problems with highly variable and rotating convective flow, indicate that the convergence is grid independent. It deteriorates moderately as the convection becomes increasingly dominating, but the convergence factor remains uniformly bounded. This conclusion is supported for both uniform and some non‐uniform (stretched) grids. Copyright © 2000 John Wiley & Sons, Ltd.     

Two positivity preserving flux limited,second‐order numerical methods for a haptotaxis model

Two numerical methods for a one‐dimensional haptotaxis model, which exploit the use of van Leer flux limiter, are developed and analyzed. Sufficient conditions time step size and flux limiting are given for such formulation to ensure the non‐negativity of the discrete solution and second‐order accuracy in space. Another advantage is that we avoid solving large nonlinear systems of algebraic equations. The discrete preservation of total conservation of cell density, concentration, and logarithmic density is also verified for the numerical solution. Numerical results concerning accuracy, convergence rate, positivity, and conservation properties are presented and discussed. Similar approach could be applied efficiently in the corresponding two‐ and three‐dimensional problems. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Optimal synchronization of two different in‐commensurate fractional‐order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional‐order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional‐order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. © 2016 Wiley Periodicals, Inc. Complexity 21: 401–416, 2016     

THE FAO PRECAUTIONARY APPROACH AFTER ALMOST 10 YEARS: HAVE WE PROGRESSED TOWARDS IMPLEMENTING SIMULATION‐TESTED FEEDBACK‐CONTROL MANAGEMENT SYSTEMS FOR FISHERIES MANAGEMENT?

ABSTRACT. It is almost ten years since the FAO Technical Consultation on the Precautionary Approach to Capture Fisheries took place in Lysekil, Sweden. One outcome from this Technical Consultation was a set of guidelines on the precautionary approach to capture fisheries and species introductions. These guidelines include the need to incorporate harvest control rules in management plans. Harvest control rules should specify what action is to be taken when specified deviations from the operational targets and constraints are observed. The specification should include minimum data requirements for the types of assessment methods to be used for decision‐making. Combinations of harvest control rules, assessment methods and data collection schemes are referred to as management procedures. It is now well‐recognized that using management procedures is likely to lead to improved conservation of fishery resources, and that they should be evaluated to assess whether they are likely to achieve the goals for fishery management given the types of uncertainties that are likely to frustrate this venture. In general, evaluation of management procedures has been based on simulation modeling. This paper reviews the progress that has been made in various fisheries jurisdictions in terms of implementing management procedures, and why and where it has proved difficult or even impossible to implement management procedures.     

Time decay rate for two 3D magnetohydrodynamics‐ α models

We consider the long time behavior of solutions to the magnetohydrodynamics‐ α model in three spatial dimensions. Time decay rate in L²‐norm of the solution is obtained. Similar results for a generalized Leray‐ α‐magnetohydrodynamics model are also established. As a by‐product, an optimal time decay rate for the Navier–Stokes‐ α model is achieved. Copyright © 2013 John Wiley & Sons, Ltd.     

WHEN TO CUT A STAND OF TREES?

This paper revisits the debate over the economic optimality of different timber harvest rules. The traditional Faustmann and Maximum Sustained Yield rotation determinations are confined to a deterministic world. Once stochasticity is introduced into the model formulation and the additional rental and management costs due to postponement of harvest are taken into account, we find that the optimal stopping time becomes random and varies in response to changes in the underlying price and growth processes. As a result, this stochastic optimal stopping time is bounded by the Faustmann cutting age from below if variabilities of the stochastic processes diminish to zero, but not necessarily by the MSY rotation from above.     

An optimal burn‐in policy based on a degradation model

Burn‐in tests help manufacturers detect defective items and remove them before being sold to customers. In a competitive marketplace, cost is a major consideration and not employing a burn‐in test may result in higher and needless expenses. With this in mind, we consider degradation‐based burn‐in tests in which the degradation path follows a Wiener process and weak items are identified when the process crosses a piecewise linear function. We also study linear functions as a special case of such a piecewise linear barrier. Within this setup, we apply a cost model to determine the optimal burn‐in test. Finally, we discuss an illustrative example using GaAs laser degradation data and present an optimal burn‐in test for it.     

Mean‐variance hedging with basis risk

Basis risk arises in a number of financial and insurance risk management problems when the hedging assets do not perfectly match the underlying asset in a hedging program. Notable examples in insurance include the hedging for longevity risks, weather index–based insurance products, variable annuities, etc. In the presence of basis risk, a perfect hedging is impossible, and in this paper, we adopt a mean‐variance criterion to strike a balance between the expected hedging error and its variability. Under a time‐dependent diffusion model setup, explicit optimal solutions are derived for the hedging target being either a European option or a forward contract. The solutions are obtained by a delicate application of the linear quadratic control theory, the method of backward stochastic differential equation, and Malliavin calculus. A numerical example is presented to illustrate our theoretical results and their interesting implications.