Designing marine protected areas for migrating fish stocks

This paper extends an earlier analysis and presents an investigation of how migration rates affect the performance of various types of management regimes with respect to economic yield and conservation benefits. Particular emphasis is placed on evaluating the geometric design of marine protected areas (MPAs). Earlier results have shown that MPAs are only likely to provide significant benefits when they are used in conjunction with direct catch or effort controls, unless they are quite large and cover most of the resource in question. Conversely, catch and effort controls are far more effective when protected areas are included in the management regime as a buffer against uncertainty. Dispersal of reproduction (recruitment) to other areas is an important expected benefit of protected areas, but such dispersal increases the variability of the effects of the area protection. If fishing mortality rates outside of the protected area are not controlled then dispersal can result in nullifying some of the benefits of the protected area. Similarly, adult migration increases the variability in the results when an area is protected and critically depends upon an overall control of fishing mortality outside the area. For both dispersal and migration separately or in combination, however, there are clear benefits to using MPAs in conjunction with catch or effort controls. These benefits are expressed in terms of long-term yield and recovery probabilities. In addition, short-term yield declines relatively slowly with increasing area protected. Design of the protected areas is seen to be important since using contiguous areas provide greater protection against overfishing than protected areas in isolation.     

Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis

Tumor anti-angiogenesis is a cancer treatment approach that aims at preventing the primary tumor from developing its own vascular network needed for further growth. In this paper the problem of how to schedule an a priori given amount of angiogenic inhibitors in order to minimize the tumor volume is considered for three related mathematical formulations of a biologically validated model developed by Hahnfeldt et al. [1999. Tumor development under angiogenic signalling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 59, 4770-4775]. Easily implementable piecewise constant protocols are compared with the mathematically optimal solutions. It is shown that a constant dosage protocol with rate given by the averaged optimal control is an excellent suboptimal protocol for the original model that achieves tumor values that lie within 1% of the theoretically optimal values. It is also observed that the averaged optimal dose is decreasing as a function of the initial tumor volume.     

Optimal harvesting of a logistic population in an environment with stochastic jumps

Dynamic programming is employed to examine the effects of large, sudden changes in population size on the optimal harvest strategy of an exploited resource population. These changes are either adverse or favorable and are assumed to occur at times of events of a Poisson process. The amplitude of these jumps is assumed to be density independent. In between the jumps the population is assumed to grow logistically. The Bellman equation for the optimal discounted present value is solved numerically and the optimal feedback control computed for the random jump model. The results are compared to the corresponding results for the quasi-deterministic approximation. In addition, the sensitivity of the results to the discount rate, the total jump rate and the quadratic cost factor is investigated. The optimal results are most strongly sensitive to the rate of stochastic jumps and to the quadratic cost factor to a lesser extent when the deterministic bioeconomic parameters are taken from aggregate antarctic pelagic whaling data.Research supported in part by the National Science Foundation under grants MCS 81-01698 and MCS 83-00562.     

Coexistence properties of some predator-prey systems under constant rate harvesting and stocking

The global behaviour of a class of predator-prey systems, modelled by a pair of non-linear ordinary differential equations, under constant rate harvesting and/or stocking of both species, is presented. Theoretically possible structures and transitions are developed and validated by computer simulations. The results are presented as transition loci in the F-G (prey harvest rate-predator harvest rate) plane.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and in part by NSERC of Canada, Grant No. 67-3138The authors wish to thank Mr. Al MacKenzie of the Department of Electrical Engineering, University of British Columbia, for preparing the figures in this paper.     

Fisheries management in practice: review of 13 commercially important fish stocks

This paper comparatively reviews several commercially important fish stocks, their state and their management in various regions of the world including Japanese anchovy, Bay of Biscay anchovy, North Sea sandeel, North Sea herring, Icelandic cod, Barents Sea cod, South African cape hakes, sockeye salmon, chinook salmon, southern bluefin tuna, Pacific halibut, Greenland halibut and Patagonian toothfish. The reviewed fish stocks are systemized in three categories: (1) stock properties and status; (2) management structure and objectives; and (3) management advice. We gather evidence to outline qualities of management regimes that are recommended and highlight those that most often fail. Robust management, biological limits (reference points), implementation and consensus are critical points that separate successful and unsuccessful management regimes. We evaluate each fish stock’s management performance relative to its management objectives and current conservation issues. Furthermore, we point out the importance of stakeholder involvement in fisheries management as well as the problems that international fisheries commissions face through examples from the case studies. Management successes tended to be single-nation and single-stock fisheries with capacity control and clear stakeholder involvement. Fisheries with fleet overcapacity, unclear objectives and illegal activity characterized the case studies with management problems.     

一类分数阶微分方程的最优控制

主要研究了一类分数阶微分方程的最优控制问题.通过Oustaloup迭代逼近,可以将分数阶微分算子在频率域范围内进行近似.那么原先的分数阶微分方程就转换为一般的常微分方程.利用这个关系,可以得到关于分数阶微分方程的两个定理和一个引理.最后给出一个例子说明该方法的有效性.     

The consequences of balanced harvesting of fish communities

Balanced harvesting, where species or individuals are exploited in accordance with their productivity, has been proposed as a way to minimize the effects of fishing on marine fish communities and ecosystems. This calls for a thorough examination of the consequences balanced harvesting has on fish community structure and yield. We use a size- and trait-based model that resolves individual interactions through competition and predation to compare balanced harvesting with traditional selective harvesting, which protects juvenile fish from fishing. Four different exploitation patterns, generated by combining selective or unselective harvesting with balanced or unbalanced fishing, are compared. We find that unselective balanced fishing, where individuals are exploited in proportion to their productivity, produces a slightly larger total maximum sustainable yield than the other exploitation patterns and, for a given yield, the least change in the relative biomass composition of the fish community. Because fishing reduces competition, predation and cannibalism within the community, the total maximum sustainable yield is achieved at high exploitation rates. The yield from unselective balanced fishing is dominated by small individuals, whereas selective fishing produces a much higher proportion of large individuals in the yield. Although unselective balanced fishing is predicted to produce the highest total maximum sustainable yield and the lowest impact on trophic structure, it is effectively a fishery predominantly targeting small forage fish.     

污染环境中可再生资源的最优收获问题

探讨了污染环境下双种群的最优收获问题．利用Pontryagin极大值原理得到一种最优分配方案——处理资源种群体内毒素的努力度与收获资源种群的努力度的分配方案,使经营者的经济收入达到最大,同时也得到次最优均衡解。     

具空间扩散的时变种群系统的最优捕获控制

讨论了一类时变种群扩散系统的最优捕获控制的非线性问题,证明了最优捕获控制的存在性,并给出了控制u（t,x）∈Uad为最优的必要条件和最优性组．     

Optimal control of treatment in a mathematical model of chronic myelogenous leukemia

We consider a mathematical model of drug therapy for chronic myelogenous leukemia for an individual patient over a fixed time horizon. The disease dynamics are given by a system of ordinary differential equations that describe the interaction between naive T cells, effector T cells and leukemic cancer cells in a hypothetical patient. We introduce two drug therapies into this model, one a targeted therapy, and the other a broad cytotoxic therapy. Our goal is to find treatment regimens that minimize the cancer cell count and the deleterious effects of the drugs for a given patient. We examine the control setting analytically, and include numerical solutions to illustrate the optimal regimens under various assumptions.     

Reduction of predation risk under the cover of darkness: Avoidance responses of mayfly larvae to a benthic fish

Summary Mayfly larvae of Paraleptophlebia heteronea (McDunnough) had two antipredator responses to a nocturnal fish predator (Rhinichthys cataractae (Valenciennes)): flight into the drift and retreat into interstitial crevices. Drift rates of Paraleptophlebia abruptly increased by 30 fold when fish were actively foraging in the laboratory streams but, even before fish were removed, drift began returning to control levels because larvae settled to the substrate and moved to areas of low risk beneath stones. This drifting response was used as an immediate escape behavior which likely decreases risk of capture from predators which forage actively at night. Surprisingly, drift most often occurred before contact between predator and prey, and we suggest that in darkness this mayfly may use hydrodynamic pressure waves for predator detection, rather than chemical cues, since fish forage in an upstream direction. Although drifting may represent a cost to mayfly larvae in terms of relocation to a new foraging area with unknown food resources, the immediate mortality risk probably out-weighs the importance of staying within a profitable food patch because larvae can survive starvation for at least 2 d. In addition to drifting, mayflies retreated from upper, exposed substrate surfaces to concealed interstitial crevices immediately after a predator encounter, or subsequent to resettlement on the substrate after predator-induced drift. A latency period was associated with this response and mayflies remained in these concealed locations for at least 3 h after dace foraging ceased. Because this mayfly feeds at night and food levels are significantly lower in field refugia under stones, relative to exposed stone surfaces, predator avoidance activity may limit foraging time and, ultimately, reduce the food intake of this stream mayfly.     

Optimal Control in a Model of Dendritic Cell Transfection Cancer Immunotherapy

We construct a population dynamics model of the competition among immune system cells and generic tumor cells. Then, we apply the theory of optimal control to find the optimal schedule of injection of autologous dendritic cells used as immunotherapeutic agent.The optimization method works for a general ODE system and can be applied to find the optimal schedule in a variety of medical treatments that have been described by a mathematical model.     

Optimization of harvesting age in an integral age-dependent model of population dynamics

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of the problem is provided. The model shows the presence of a zero-investment period. A preliminary asymptotic analysis indicates possible turnpike properties of the optimal harvesting age. Biological interpretation of all results is provided.     

Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock’s direct multiple shooting method.In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.     

Repeated applications of a transdermal patch: analytical solution and optimal control of the delivery rate

The integral transform technique was implemented to solve a mathematical model developed for percutaneous drug absorption. The model included repeated application and removal of a patch from the skin. Fick's second law of diffusion was used to study the transport of a medicinal agent through the vehicle and subsequent penetration into the stratum corneum. Eigenmodes and eigenvalues were computed and introduced into an inversion formula to estimate the delivery rate and the amount of drug in the vehicle and the skin. A dynamic programming algorithm calculated the optimal doses necessary to achieve a desired transdermal flux. The analytical method predicted profiles that were in close agreement with published numerical solutions and provided an automated strategy to perform therapeutic drug monitoring and control.     

Optimal control of the chemotherapy of HIV

 Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The control represents the percentage of effect the chemotherapy has on the viral production. Using an objective function based on a combination of maximizing benefit based on T cell counts and minimizing the systemic cost of chemotherapy (based on high drug dose/strength), we solve for the optimal control in the optimality system composed of four ordinary differential equations and four adjoint ordinary differential equations. Received 5 July 1995; received in revised form 3 June 1996     

Antagonistic control of microbial pathogens under iron limitations by siderophore producing bacteria in a chemostat setup

Certain bacteria develop iron chelation mechanisms that allow them to scavenge dissolved iron from the environment and to make it unavailable to competitors. This is achieved by producing siderophores that bind the iron which is later liberated internally in the cell. Under conditions of iron limitation, siderophore producing bacteria have therefore an antagonistic growth advantage over other species. This has been observed in particular in agricultural and aquacultural systems, as well as in food microbiology. We investigate here the possibility of a probiotic biocontrol strategy to eradicate a well established, often pathogenic, non-chelating population by supplementing the system with generally regarded as safe siderophore producing bacteria. Set in a chemostat setup, our modeling and simulation studies suggest that this is indeed possible in a finite time treatment.