Optimal Harvesting in Stream Networks: Maximizing Biomass and Yield

We analyze the long-term behavior of interacting populations which can be controlled through harvesting. The dynamics is assumed to be discrete in time and stochastic due to the effect of environmental fluctuations. We present powerful extinction and coexistence criteria when there are one or two interacting species. We then use these tools in order to see when harvesting leads to extinction or persistence of species, as well as what the optimal harvesting strategies, which maximize the expected long-term yield, look like. For single species systems, we show under certain conditions that the optimal harvesting strategy is of bang-bang type: there is a threshold under which there is no harvesting, while everything above this threshold gets harvested. We are also able to show that stochastic environmental fluctuations will, in most cases, force the expected harvesting yield to be lower than the deterministic maximal sustainable yield. The second part of the paper is concerned with the analysis of ecosystems that have two interacting species which can be harvested. In particular, we carefully study predator–prey and competitive Ricker models. We are able to analytically identify the regions in parameter space where the species coexist, one species persists and the other one goes extinct, as well as when there is bistability. We look at how one can find the optimal proportional harvesting strategy. If the system is of predator–prey type, the optimal proportional harvesting strategy is, depending on the interaction parameters and the price of predators relative to prey, either to harvest the predator to extinction and maximize the asymptotic yield of the prey or to not harvest the prey and to maximize the asymptotic harvesting yield of the predators. If the system is competitive, in certain instances it is optimal to drive one species extinct and to harvest the other one. In other cases, it is best to let the two species coexist and harvest both species while maintaining coexistence. In the setting of the competitive Ricker model, we show that if one competitor is dominant and pushes the other species to extinction, the harvesting of the dominant species can lead to coexistence.

A simple model for seaweed growth and optimal harvesting strategy

An optimization model was developed to determine the optimal harvesting strategy needed for uneven-aged mixed-species stands in the Changbai Mountain region of northeast of China. The model takes into account four variables including residual basal area (RBA), the diameter of the largest tree, harvest cycle and a constant representing the ratio of the number of trees in a given diameter class to those in the next larger diameter class (‘q’). According to model simulations, under the objective of maximizing net revenue, the optimal harvesting strategy is defined when the residual basal area equals to 19 m2 ha−1, the diameter of the largest tree equals to 44 cm, q 1.3 and the harvest cycle equals to 20 years. If the objective is to maximize the total volume yield, the optimal harvesting strategy is defined when RBA equals to 13 m2 ha−1, the diameter of the largest tree equals to 36 cm and the constant ‘q’ equals to 1.9 and the harvest cycle equals to 15 years.

Olive oil has been favored as high-quality edible oil because it contains balanced fatty acids (FAs) and high levels of minor components. The contents of FAs and minor components are variable in olive fruits of different color at harvest time, which render it difficult to determine the optimal harvest strategy for olive oil producing. Here, we combined metabolome, Pacbio Iso-seq, and Illumina RNA-seq transcriptome to investigate the association between metabolites and gene expression of olive fruits at harvest time. A total of 34 FAs, 12 minor components, and 181 other metabolites (including organic acids, polyols, amino acids, and sugars) were identified in this study. Moreover, we proposed optimal olive harvesting strategy models based on different production purposes. In addition, we used the combined Pacbio Iso-seq and Illumina RNA-seq gene expression data to identify genes related to the biosynthetic pathways of hydroxytyrosol and oleuropein. These data lay the foundation for future investigations of olive fruit metabolism and gene expression patterns, and provide a method to obtain olive harvesting strategies for different production purposes.

Optimal harvest strategy based on a discrete age-structured model with monthly fishing effort for chub mackerel, Scomber japonicus, in South Korea

Summary Large carnivores are currently recolonizing areas where they have been extinct for decades. This poses considerable challenges for wildlife managers, partly because the optimal harvesting strategies of prey populations may be affected. If the carnivores in such areas are under strict management control (as in Scandinavia), the predator will not show a numerical response to changes in prey density. Consequently, the density of prey is mainly determined by the vital rates of the prey population and the predation pressure. In this study we modelled the optimal harvesting strategy for a prey population in which there was no numerical response by the predator. Our model is an age‐structured deterministic matrix model system. Optimal harvesting strategies are determined, measuring yield either as number of animals harvested or as mass of meat. First, using a moose population in Hedmark in south‐eastern Norway as a case study, we demonstrate that continuing to harvest at the rates used prior to wolf recolonization will lead to a decline in the moose population. Secondly, harvesting quotas are specified by age and sex, usually with a high proportion of calves. Although wolves mainly kill juvenile moose (calves and yearlings), the relationship between harvest composition and yield is not affected by predation. Both in the presence and absence of predation, a high proportion of calves in the harvest gives the highest yield measured as the number of animals harvested, whereas a high proportion of adults maximizes the yield measured in terms of meat. Furthermore, a female‐biased sex structure in the population gives a higher yield in both the presence and absence of the predator. Synthesis and applications. We have shown that managers facing the new challenges presented by recolonizing populations of large predators such as wolves should reduce the size of harvest quotas in order to avoid decreases in prey populations. However, the general relationship between the harvesting strategy and yield maximization is not affected by wolf predation. The harvest yield from cervid populations is often important to local economies, and moose is the single most important game species in Scandinavia. It is therefore important to implement optimal harvesting strategies under these new conditions in order to prevent an unnecessary loss of yield, and success in this task may in turn affect local attitudes towards large carnivores.

Water Distribution Networks (WDNs) represent a noteworthy field for possible implementation of Small Hydropower (SHP), by replacing Pressure Reduction Valves (PRV) with turbomachines, in particular Pump as Turbines (PaTs), to control and regulate the pressure, while harvesting energy otherwise wasted. Different models were developed to predict the performance and select the positioning of the PaTs for the maximum energy recovery but most of them neglect practical aspect such as: power grid limitations and optimal harvesting strategy. In this framework, we intend to propose a new method to select a PaT, defining its optimal working point, by introducing an energy exploitation coefficient. The proposed methodology is based on the experimental results of a real PaT tested in the high capacity hydraulic laboratory at Polytechnic University of Bari. Firstly, the selected commercial centrifugal pump was tested in both pump and turbine modes. Then, three different approaches, for the Best Efficiency Point (BEP) selection, are described and compared in terms of energy exploitation and capacity factor for a WDN. The first consists of selecting the BEP at the average flow rate, the second one considers the probability distribution of the flow rate and the corresponding available hydraulic energy, whereas the latter is based on the highest energy harvesting. By applying energy production, economic and environmental analyses, the new proposed methodology, based on the third approach, shows a remarkable advantage in terms of exploited energy. Indeed a remarkable 60% energy recovery is achieved with 334 ton CO2/year avoided. Furthermore, the impact of the electrical motor on the maximum power generation (cut-off) is considered. Eventually, useful insights for the future PaT selection and installation are discussed.

A stabilizing harvesting strategy for an uncertain model of an ecological system

This work focuses on optimal harvesting problems for a stochastic competitive ecosystem subject to Levy noise. A verification theorem for corresponding harvesting strategy is established, which offers sufficient conditions for deriving an optimal harvesting strategy and an upper bound of the value function. For a given instantaneous marginal yields function, a concrete upper bound of value function is constructed by applying the verification theorem obtained in this paper. Meanwhile, the monotonicity of value function is investigated. Also, an e-optimal harvesting strategy is designed to find an approximate optimal harvesting strategy for those harvesting problems with no exact optimal harvesting strategy. Finally, by choosing appropriately Markov decision process defined on a discrete state space, a computational method for an optimal harvesting strategy is designed and a concrete example is also given to show the implementation of the algorithm.

Optimal sustainable harvesting of populations in random environments

Harvesting a population sustainably as a resource is a common problem in wildlife and fisheries management. In a typical situation population size and other state variables are monitored at fixed intervals and the resulting estimates are used to determine a quota or harvest effort. Such decisions must be made in the face of a range of uncertainties: environmental variation, an imperfect ability to observe state variables, an imperfect ability to implement management decisions and imperfect knowledge about how the population fluctuates dynamically in response to management actions. In this thesis I explore the effect of these uncertainties on the optimal management of wild populations. In chapters 2 and 3, populations with significant age or stage structure are examined. When individuals of different maturity within a population have significantly different life history traits, the structure of the population and of the harvest taken can have a large effect on the growth rate of the population. In chapter 2 a simple matrix model is used to make observations about the role of demographic structure when the objective is to maintain the population below its carrying capacity. Uncertainty in the structure of the population and the manager's ability to select the harvest structure complicate optimal harvesting. In chapter 3 plausible models are developed for the maintenance of the Atlantic population of Canada geese (Branta canadensis) within acceptable population bounds, given uncertainty about the strength of density dependent population regulation and the limited ability of managers to achieve large harvests. Stochastic dynamic programming is used to determine the optimal harvest strategy under each of the plausible models. It is found that the target long-term population size depends critically on the strength of density dependence. Under the density-independent model, limits to harvest also influence the target long-term population size. Chapter 4 explores the theory of adaptive management. In adaptive management we seek the optimal harvest decision in the presence of model uncertainty. Plausible models are weighted according to the amount of evidence currently supporting them. The optimal harvest decision is obtained by weighting the expected returns under each model. When the population is monitored subsequent to harvesting, the evidence supporting each model can be re-evaluated. In this way the model best describing the system dynamics can be learnt over time (passive adaptive management). Particular harvest decisions may accelerate learning and provide better management in the long term. However these actions are often perceived as risky and so short-term losses must be balanced by long-term benefits (active adaptive management). To test these ideas, a simple population model with an uncertain parameter is constructed. Fixed, passive adaptive and active adaptive harvest strategies are developed using stochastic dynamic programming. It is found that the passive adaptive strategy is `certainty-equivalent', meaning that the current best estimate of the uncertain parameter is used as if it were the true parameter value. Over very long time horizons, the active adaptive strategy probes for information but in the short-term it is actually more precautionary than the certainty-equivalent strategy. The passive and active adaptive strategies perform similarly well in maximising harvest, and both outperform fixed non- adaptive strategies. Two different sets of plausible models produce consistent results, leading to the conclusion that it is important to incorporate model uncertainty, but the specific approach does not critically affect the results. In chapter 5 the problem of optimal adaptive monitoring is considered. The most common approach to harvest management is to use the same monitoring effort at regular intervals to estimate state variables. This approach neglects the large costs often involved in population monitoring, assuming that the level of accuracy achieved is both necessary and sufficient to make the appropriate harvest decision. I take an alternative approach, combining the costs of monitoring and the expected benefits for management in a single framework to determine the level of monitoring accuracy required. Monitoring effort becomes a state-dependent decision at each time interval, determined by prior information about the state variables. This approach is demonstrated using data for a red kangaroo (Macropus rufus) population in South Australia. This document is not a comprehensive treatment of optimal harvesting under uncertainty. However it does indicate the ways in which uncertainty complicates the harvest of wildlife, and its potential effect on optimal harvesting and monitoring decisions.

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction-instead the harvested population converges to a unique invariant probability measure. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold [Formula: see text] such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is [Formula: see text] and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.

This paper investigates the optimal harvesting strategy for a single species living in random environments whose growth is given by a regime-switching diffusion. Harvesting acts as a (stochastic) control on the size of the population. The objective is to find a harvesting strategy which maximizes the expected total discounted income from harvesting {\em up to the time of extinction} of the species; the income rate is allowed to be state- and environment-dependent. This is a singular stochastic control problem with both the extinction time and the optimal harvesting policy depending on the initial condition. One aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may significantly alter the extinction time when using the same harvesting policy. Consequently, one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fixed time horizon. This paper introduces a new sufficient condition under which the continuity of the value function for the regime-switching model is established. Further, it is shown that the value function is a viscosity solution of a coupled system of quasi-variational inequalities. The paper also establishes a verification theorem and, based on this theorem, an $\varepsilon$-optimal harvesting strategy is constructed under certain conditions on the model. Two examples are analyzed in detail.

Understanding variation in cellular growth rates among cells in tumors is crucial for predicting cancer progression and interpreting tumor-derived genetic data. Advances in lineage tracing technologies enable the reconstruction of high-resolution, single-cell phylogenies of cancer cell populations, but methods to detect cellular growth rate differences on these phylogenies remain limited. Tree balance statistics offer a way forward, but it is unknown if and how these statistics are distorted when applied to phylogenetic reconstructions built from lineage tracing data, and if these distortions limit the utility of tree balance statistics to distinguish between evolutionary scenarios characterized by variable or homogeneous cellular growth rates. Here, we examined two tree balance statistics, and the Sackin index, and benchmarked their performance in distinguishing lineage tracing trees derived from populations with and without variable cellular growth rates. We found that when tumor population sizes and lineage tracing editing rates are approximately known and in favorable ranges, detects departures from homogenous growth rates just as well on lineage tracing trees as on true genealogical trees, while the Sackin index loses most of its power even under the most favorable conditions. We applied our -based test to data derived from cancer lineage tracing experiments and found widespread signals of growth rate heterogeneity in murine autochthonous lung cancers, and lung and PDAC xenograft experiments in mice. Our results demonstrate the potential and challenges of tree balance statistics in analyzing growth dynamics in lineage tracing data.

Ungulate population dynamics and optimization models