A Preconditioned Iterative Method for the Solution of the Biharmonic Problem

The numerical solution of the first biharmonic problem on polygonalplane domains is considered. A method is derived, which allowsan arbitrary degree of finite-element approximation and thediscretized system is solved by a fast iterative procedure.Theoretical estimates for the computational complexity are derivedand a number of test examples are examined.     

OPTIMAL HARVEST FEEDBACK RULE ACCOUNTING FOR THE FISHING‐UP EFFECT AND SIZE‐DEPENDENT PRICING

Abstract Fishing leads to truncation of a population's age and size structure. However, large‐sized fish are usually more valuable per unit weight than small ones. Nevertheless, these size‐related factors have mostly been ignored in bioeconomic modeling. Here, we present a simple extension to the Gordon–Schaefer model that accounts for variations in mean individual catch weight, and derive the feedback rule for optimal harvest in this setting. As the Gordon–Schaefer model has no population structure, size effects have to be accounted for indirectly. Here we assume a simple negative relationship between fishing effort and mean individual weight, and a positive relationship between mean catch weight and price. The aim is to emulate alterations of size structure in fish populations due to fishing and the influence of size on price per weight unit and eventually, net revenues. This demonstrates, on a general level, how such size‐dependent effects change the patterns of optimal harvest paths and sustainable revenue in single fish stocks. The model shows clear shifts toward lower levels of optimal effort and yield compared to classical models without size effects. This suggests that ignoring body size could lead to misleading assumptions and policies, potentially causing rent dissipation and suboptimal utilization of renewable resources.     

INTERACTION BETWEEN TYROSINE AND DIVALENT SULFUR IN FLUORESCENCE QUENCHING AND IN THE PHOTOCHEMISTRY OF RIBONUCLEASE

Abstract— –Ribonuclease is inactivated in aqueous solution by u.v. light through different mechanisms according to whether divalent sulfur or aromatic amino acids are the primary light absorbers. At 284 nm, absorbed mainly by tyrosine, the presence of O₂ inhibits photoinactivation and H₂S formation, but does less so at 254 or 313 nm. Based on data with model substances containing disulfide groups a mechanism is indicated in which excited tyrosine is quenched through electron transfer to adjacent divalent sulfur within the protein. Disulfide compounds are shown to be very efficient quenchers of tyrosine fluorescence.     

Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

Discretization of autonomous ordinary differential equationsby numerical methods might, for certain step sizes, generatesolution sequences not corresponding to the underlying flow—so-called‘spurious solutions’ or ‘ghost solutions’.In this paper we explain this phenomenon for the case of explicitRunge-Kutta methods by application of bifurcation theory fordiscrete dynamical systems. An important tool in our analysisis the domain of absolute stability, resulting from the applicationof the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inheritedby the difference scheme induced by the numerical method whilethe stability type of these inherited genuine fixed points iscompletely determined by the method's domain of absolute stability.We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the correspondinglimit step size. Moreover, we show in which way the bifurcationsoccurring at the limit step size are connected to the valuesof the stability function on the boundary of the domain of absolutestability, where we pay special attention to bifurcations leadingto spurious solutions. In order to explain a certain kind ofspurious fixed points which are not connected to the set ofgenuine fixed points, we interprete the domain of absolute stabilityas a Mandeibrot set and generalize this approach to nonlinearproblems.     

A SIMPLIFIED FEEDBACK APPROACH TO OPTIMAL RESOURCE MANAGEMENT

ABSTRACT. Optimal exploitation of a renewable capital stock is derived as a feedback rule for a general dynamic optimization problem with a single resource. By a feedback rule is meant that optimal exploitation is given as an explicit function of the capital stock. The value of the simplified approach will be appreciated by all who have tried to determine the separatrix solution numerically and experienced problems. The method described here can be applied to fisheries management, animal stock conservation and conservation of the environment in general. The operationality of the method outlined here is illustrated by two simple examples, one related to fisheries management and one related to pollution control.     

OPTIMAL ENVIRONMENTAL TAXES: EFFECTS OF POLLUTION DECAY AND CONSUMER AWARENESS

Abstract The effects of nonlinear decay and consumer preferences are analyzed in a setting where optimal extraction of nonrenewable resources is combined with stock externalities. The control is exercised via a corrective tax and the time horizon is divided into two periods: an initial phase with extraction and a terminal phase without extraction. The time horizon with extraction is determined endogenously. The model does not assume separability of the objective function. The purpose here is to demonstrate that relatively simple deviations from the standard assumptions, such as linear decay and no consumer awareness, may have large effects. Sensitivity analyses indicate large differences in the optimal extraction period, the total level of extraction and cumulative emissions depending on the form of the decay function and the presence of consumers' awareness for the environment.     

WHY INCREASED UNCERTAINTY MAY LEAD TO MORE RISKY BEHAVIOR

The purpose of this article is to investigate circumstances under which it may be optimal to deliberately harvest a fish stock to extinction applying a stochastic surplus growth model. It is known from the literature that deliberate extinction may result when there is critical depensation or when the discount rate is high compared to the intrinsic growth rate. Here it is shown that deliberate extinction may also be optimal when the degree of stochasticitry is high even with zero discounting. A high degree of stochasticity may have the same effect as critical depensation even though it is not present in the biological model. In other words, high uncertainty, instead of leading to more conservative harvesting as is usually expected, in this model result in more aggressive harvesting and more risky behavior. The main message is therefore always to try to keep the stock well above any critical limit.     

INFERRING A BIOPOLITICAL CONSENSUS VIEW OF STOCHASTIC DYNAMICS FOR MANAGEMENT OF A TRANSBOUNDARY FISHERY

ABSTRACT. Management of trans‐boundary fisheries is a complicated problem with biological, legal, economic and political implications. We propose a simple stochastic differential‐equation model to describe a biopolitical consensus view of fish stock dynamics. Estimates of the drift and diffusion terms of three stochastic differential equations are obtained using data from the southern bluefin tuna (SBT) fishery with a method based on the Kolmogorov‐Smirnov statistic. We refer to these estimated equations as alternative biopolitical consensus views of SBT stock dynamics. Each of these is used to generate a time series of optimal harvest that achieves the objective of maximizing the present value of expected fishery returns. These time series of optimal harvests are then compared to actual harvests for the period 1981 1997.