The effect of optimal harvesting on the dynamics of size and genetic composition of a two-age population

The effect of optimal stationary harvesting at a constant harvest rate on the dynamics of a two-age population is considered. It has been shown analytically that harvesting a fixed rate of the population size of only one age cohort is optimal. As has been observed, the maximum of revenue function is unattainable in the case of concurrent harvesting of both age cohorts. It has been demonstrated that the direction of natural selection does not explicitly change when unselectively harvesting individuals; however, the adaptive genetic diversity of an unharvested population can be lost due to harvesting.     

Chronic Wasting Disease: Transmission Mechanisms and the Possibility of Harvest Management

We develop a model of CWD management by nonselective deer harvest, currently the most feasible approach available for managing CWD in wild populations. We use the model to explore the effects of 6 common harvest strategies on disease prevalence and to identify potential optimal harvest policies for reducing disease prevalence without population collapse. The model includes 4 deer categories (juveniles, adult females, younger adult males, older adult males) that may be harvested at different rates, a food-based carrying capacity, which influences juvenile survival but not adult reproduction or survival, and seasonal force of infection terms for each deer category under differing frequency-dependent transmission dynamics resulting from environmental and direct contact mechanisms. Numerical experiments show that the interval of transmission coefficients β where the disease can be controlled is generally narrow and efficiency of a harvest policy to reduce disease prevalence depends crucially on the details of the disease transmission mechanism, in particular on the intensity of disease transmission to juveniles and the potential differences in the behavior of older and younger males that influence contact rates. Optimal harvest policy to minimize disease prevalence for each of the assumed transmission mechanisms is shown to depend on harvest intensity. Across mechanisms, a harvest that focuses on antlered deer, without distinguishing between age classes reduces disease prevalence most consistently, whereas distinguishing between young and older antlered deer produces higher uncertainty in the harvest effects on disease prevalence. Our results show that, despite uncertainties, a modelling approach can determine classes of harvest strategy that are most likely to be effective in combatting CWD.     

Effects of harvesting onAscophyllum nodosum (L.) Le Jol. (Fucales,Phaeophyta): a demographic approach

Although populations ofAscophyllum nodosum are harvested commercially, little is known about the effects on demographic vital rates (growth, reproduction, survival). This study examines the effects of harvesting season and harvesting intensity on growth, reproduction and mortality of intact fronds in four size classes and in fronds truncated by the harvest. Knowledge of size-specific vital rates was used to evaluate the response of the population to harvesting.Harvesting season and harvesting intensity did not exert a significant effect on growth. Growth in plots not subject to harvesting was less than in harvested plots. No major differences in growth, reproduction and survival between intact and severed fronds emerged. The number of fronds attaining reproduction was enhanced by increased harvesting intensity and by cutting in summer. Harvesting did not seem to induce breakage, and breakage appeared higher in the uncut plots. Most harvesting treatments did not influence survivorship and survivorship was similar among all size classes. Growth rates were inversely related to sizes of fronds.Assessment of variation across size classes yielded more accurate estimates of growth rates than those of previously used methods. Accurate size class specific-growth rates will be a useful criterion when regulating intervals between harvests. Furthermore, assessment of size-specific vital rates allows identification of the frond size classes most relevant to the preservation of resources. Because of their fast growth rates and abundance, fronds in class 1, and, to a lesser extent, class 2, are responsible for most of the population regrowth after harvest. In contrast, classes 3 and 4 contribute little to recovery. This finding provides a strong basis for a harvesting strategy that targets the largest fronds.Author for correspondence     

Harvesting in a resource dependent age structured Leslie type population model

We analyse the effect of harvesting in a resource dependent age structured population model, deriving the conditions for the existence of a stable steady state as a function of fertility coefficients, harvesting mortality and carrying capacity of the resources. Under the effect of proportional harvest, we give a sufficient condition for a population to extinguish, and we show that the magnitude of proportional harvest depends on the resources available to the population. We show that the harvesting yield can be periodic, quasi-periodic or chaotic, depending on the dynamics of the harvested population. For populations with large fertility numbers, small harvesting mortality leads to abrupt extinction, but larger harvesting mortality leads to controlled population numbers by avoiding over consumption of resources. Harvesting can be a strategy in order to stabilise periodic or quasi-periodic oscillations in the number of individuals of a population.     

Harvesting in seasonal environments

Most harvest theory is based on an assumption of a constant or stochastic environment, yet most populations experience some form of environmental seasonality. Assuming that a population follows logistic growth we investigate harvesting in seasonal environments, focusing on maximum annual yield (M.A.Y.) and population persistence under five commonly used harvest strategies. We show that the optimal strategy depends dramatically on the intrinsic growth rate of population and the magnitude of seasonality. The ordered effectiveness of these alternative harvest strategies is given for different combinations of intrinsic growth rate and seasonality. Also, for piecewise continuous-time harvest strategies (i.e., open / closed harvest, and pulse harvest) harvest timing is of crucial importance to annual yield. Optimal timing for harvests coincides with maximal rate of decline in the seasonally fluctuating carrying capacity. For large intrinsic growth rate and small environmental variability several strategies (i.e., constant exploitation rate, linear exploitation rate, and time-dependent harvest) are so effective that M.A.Y. is very close to maximum sustainable yield (M.S.Y.). M.A.Y. of pulse harvest can be even larger than M.S.Y. because in seasonal environments population size varies substantially during the course of the year and how it varies relative to the carrying capacity is what determines the value relative to optimal harvest rate. However, for populations with small intrinsic growth rate but subject to large seasonality none of these strategies is particularly effective with M.A.Y. much lower than M.S.Y. Finding an optimal harvest strategy for this case and to explore harvesting in populations that follow other growth models (e.g., involving predation or age structure) will be an interesting but challenging problem.     

Marine reserves and optimal harvesting

Advocates of no‐take marine reserves emphasize their conservation benefits. Critics counter that reserves would decrease fisheries yield. Analysis of a spatially explicit harvesting model, however, shows that no‐take marine reserves are always part of an optimal harvest designed to maximize yield. The optimal harvest generates a spatial source–sink structure with source populations placed in reserves. The sizes and locations of the optimal reserves depend on a dimensionless length parameter. For small values of this parameter, the maximum yield is obtained by placing a large reserve in the centre of the habitat. For large values of this parameter, the optimal harvesting strategy is a spatial ‘chattering control’ with infinite sequences of reserves alternating with areas of intense fishing. Such a chattering strategy would be impossible to actually implement, but in these cases an approximate yet practicable policy, utilizing a small number of reserves, can be constructed.     

Impact of unintentional selective harvesting on the population dynamics of red grouse

1.?The effect of selective exploitation of certain age, stage or sex classes (e.g., trophy hunting) on population dynamics is relatively well studied in fisheries and sexually dimorphic mammals. 2.?Harvesting of terrestrial species with no morphological differences visible between the different age and sex classes (monomorphic species) is usually assumed to be nonselective because monomorphicity makes intentionally selective harvesting pointless and impractical. But harvesting of the red grouse (Lagopus lagopus scoticus), a monomorphic species, was recently shown to be unintentionally selective. This study uses a sex- and age-specific model to explore the previously unresearched effects of unintentional harvesting selectivity. 3.?We examine the effects of selectivity on red grouse dynamics by considering models with and without selectivity. Our models include territoriality and parasitism, two mechanisms known to be important for grouse dynamics. 4.?We show that the unintentional selectivity of harvesting that occurs in red grouse decreases population yield compared with unselective harvesting at high harvest rates. Selectivity also dramatically increases extinction risk at high harvest rates. 5.?Selective harvesting strengthens the 3- to 13-year red grouse population cycle, suggesting that the selectivity of harvesting is a previously unappreciated factor contributing to the cycle. 6.?The additional extinction risk introduced by harvesting selectivity provides a quantitative justification for typically implemented 20-40% harvest rates, which are below the maximum sustainable yield that could be taken, given the observed population growth rates of red grouse. 7.?This study shows the possible broad importance of investigating in future research whether unintentionally selective harvesting occurs on other species.     

Effects of Commercial Harvesting on Population Characteristics and Rhizome Yield of Anemone altaica

Effects of Commercial Harvesting on Population Characteristics and Rhizome Yield of Anemone altaica. Commercial harvesting constitutes a direct threat to numerous non–timber forest products (NTFPs), but its ecological effects have not been well documented. Anemone altaica Fisch. ex C. A. Mey, a spring ephemeral plant found in temperate forests of Eurasia, is a traditional Chinese herb. Owing to medicinal value, its rhizomes have been harvested for commercial purposes in northwestern China for many years. This paper addresses the ecological effects of commercial harvesting on A. altaica populations under different harvest intensities. The results show that size–selective harvesting of rhizomes can increase population densities by asexual propagation. Currently, two– to three–year–old individuals derived from asexual propagation are the main targets of commercial harvesting. The increased demand in recent years has resulted in earlier and more intensive harvesting activities largely impacting the natural recovery of the harvested populations. For sustainable use of this traditional medicinal species, we recommend that a periodic harvest strategy of three to four years be adopted.     

Plucking or cutting Gelidium sesquipedale? A demographic simulation of harvest impact using a population projection matrix model.

A matrix model describes the annual dynamics of a commercial (harvested by plucking) Gelidium sesquipedale population off Cape Espichel, Portugal. Vital rates were measured from a frond population divided into size classes; annual transition probabilities among them were calculated. Transition probabilities under harvest by cutting are derived by assuming that all harvested fronds are cut to the first size class, and none are plucked. Simulations of the annual population dynamics for harvest by both plucking and cutting are used to assess which harvest strategy will optimize yields. Assuming the same efficiency for both strategies, cutting fronds to 7 cm (as mechanical harvesters do) results in a higher population growth rate ( = 1.08 to 1.35) than occurs with the plucking technique ( = 0.85). Simulations of population recovery show the number of fronds in each size class available the next harvest season will be higher when cut than plucked. This model can also optimize yields by predicting the more efficient season opening, and harvesting cutting height.     

The structure of optimal time- and age-dependent harvesting in the Lotka-McKendrik population model

The paper analyzes optimal harvesting of age-structured populations described by the Lotka-McKendrik model. It is shown that the optimal time- and age-dependent harvesting control involves only one age at natural conditions. This result leads to a new optimization problem with the time-dependent harvesting age as an unknown control. The integral Lotka model is employed to explicitly describe the time-varying age of harvesting. It is proven that in the case of the exponential discounting and infinite horizon the optimal strategy is a stationary solution with a constant harvesting age. A numeric example on optimal forest management illustrates the theoretical findings. Discussion and interpretation of the results are provided.     

Optimal harvesting of an age-structured population

Here we investigate the optimal harvesting of an age-structured population. We use the McKendrick model of population dynamics, and optimize a discounted yield on an infinite time horizon. The harvesting function is allowed to depend arbitrarily on age and time and its magnitude is unconstrained. We obtain, in addition to existence, the qualitative result that an optimal harvesting policy consists of harvesting at no more than three distinct ages.     

Optimal harvesting of prey–predator system with interval biological parameters: A bioeconomic model

The paper presents the study of one prey one predator harvesting model with imprecise biological parameters. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin’s maximum principle. Numerical examples are presented to support the proposed model.     

A Systematic Overview of Harvesting-Induced Maturation Evolution in Predator–Prey Systems with Three Different Life-History Tradeoffs

There are concerns that anthropogenic harvesting may cause phenotypic adaptive changes in exploited wild populations, in particular maturation at a smaller size and younger age. In this paper, we study the evolutionarily stable size at maturation of prey subjected to size-selective harvesting in a simple predator?Cprey model, taking into account three recognized life-history costs of early maturation, namely reduced fecundity, reduced growth, and increased mortality. Our analysis shows that harvesting large individuals favors maturation at smaller size compared to the unharvested system, independent of life-history tradeoff and the predator??s prey-size preference. In general, however, the evolutionarily stable maturation size can either increase or decrease relative to the unharvested system, depending on the harvesting regime, the life-history tradeoff, and the predator??s preferred size of prey. Furthermore, we examine how the predator population size changes in response to adaptive change in size at maturation of the prey. Surprisingly, in some situations, we find that the evolutionarily stable maturation size under harvesting is associated with an increased predator population size. This occurs, in particular, when early maturation trades off with growth rate. In total, we determine the evolutionarily stable size at maturation and associated predator population size for a total of forty-five different combinations of tradeoff, harvest regime, and predated size class.     

Optimal Strategies for Managing Wildlife Harvest Under System Change

Wildlife populations are experiencing shifting dynamics due to climate and landscape change. Management policies that fail to account for non-stationary dynamics may fail to achieve management objectives. We establish a framework for understanding optimal strategies for managing a theoretical harvested population under non-stationarity. Building from harvest theory, we develop scenarios representing changes in population growth rate () or carrying capacity () and derive time-dependent optimal harvest policies using stochastic dynamic programming. We then evaluate the cost of falsely assuming stationarity by comparing the outcomes of forward projections in which either the optimal policy or a stationary policy is applied. When declines over time, the stationary policy leads to an underharvest of the population, resulting in less harvest over the short term but leaving the population in a higher-value state. When declines over time, the stationary policy leads to overharvest, resulting in greater harvest returns in the short term but leaving the population in a lower and potentially more vulnerable state. This work demonstrates the basic properties of time-dependent harvest management and provides a framework for evaluating the many outstanding questions about optimal management strategies under climate change. Published 2021. This article is a U.S. Government work and is in the public domain in the USA.     

Optimal models for social change

By assigning coordinates to the environmental function space comprising all physical and mental stimuli, mathematical interpretations can be based on such terms as adaptability, and reactivity which relate to individuals interacting with their environment within a society. These psychometric concepts are incorporated into a framework of functional analysis, which permits the optimization of social change by maximizing the satisfaction integral through the use of variational or dynamic programming methods in conjunction with some optimal social policy. The approach provides a mathematical connection between psychology and sociology, and further demonstrates that existing forms of government are simulated by differential equations belonging to the same general class. The synthesis of new classes of functional equations describing social progress is visualized as a legitimate objective for abstract mathematical sociology.     

Modelling the Impact of Marine Reserves on a Population with Depensatory Dynamics

In this study, we use a spatially implicit, stage-structured model to evaluate marine reserve effectiveness for a fish population exhibiting depensatory (strong Allee) effects in its dynamics. We examine the stability and sensitivity of the equilibria of the modelled system with regards to key system parameters and find that for a reasonable set of parameters, populations can be protected from a collapse if a small percentage of the total area is set aside in reserves. Furthermore, the overall abundance of the population is predicted to achieve a maximum at a certain ratio \(A\) of reserve area to fished area, which depends heavily on the other system parameters such as the net export rate of fish from the marine reserves to the fished areas. This finding runs contrary to the contested “equivalence at best” result when comparing fishery management through traditional catch or effort control and management through marine reserves. Lastly, we analyse the problem from a bioeconomics perspective by computing the optimal harvesting policy using Pontryagin’s Maximum Principle, which suggests that the value for \(A\) which maximizes the optimal equilibrium fishery yield also maximizes population abundance when the cost per unit harvest is constant, but can increase substantially when the cost per unit harvest increases with the area being harvested.     

Bio-economics of a renewable resource in a seasonally varying environment

In this paper we study the bio-economics of a renewable resource with governing dynamics described by two distinct growth functions (viz., logistic and Gompertz growth functions) in a seasonally varying environment. Seasonality is introduced into the system by taking the involved ecological parameters to be periodic. In this work, we establish a procedure to obtain the optimal path and compute the optimal effort policy which maximizes the net revenue to the harvester for a fairly general optimal control problem and apply this procedure to the considered models to derive some important conclusions. These problems are solved on the infinite horizon. We find that, for both the models, the optimal harvest policy and the corresponding optimal path are periodic after a finite time. We also obtain optimal solution, a suboptimal harvesting policy and the corresponding suboptimal approach path to reach this optimal solution. The key results are illustrated using numerical simulations and we compare the revenues to the harvester along the optimal and suboptimal paths. The general procedure developed in this work, for obtaining the optimal effort policy and the optimal path, has wider applicability.     

The Beverton-Holt model with periodic and conditional harvesting

In this theoretical study, we investigate the effect of different harvesting strategies on the discrete Beverton-Holt model in a deterministic environment. In particular, we make a comparison between the constant, periodic and conditional harvesting strategies. We find that for large initial populations, constant harvest is more beneficial to both the population and the maximum sustainable yield. However, periodic harvest has a short-term advantage when the initial population is low, and conditional harvest has the advantage of lowering the risk of depletion or extinction. Also, we investigate the periodic character under each strategy and show that periodic harvesting drives population cycles to be multiples (period-wise) of the harvesting period.