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.     

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.     

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.     

Evolutionary responses to harvesting in ungulates

1. We investigate the evolutionary responses to harvesting in ungulates using a state-dependent, stochastic, density-dependent individual-based model of red deer Cervus elaphus (L.) females subject to different harvesting regimes. 2. The population's mean weight at first reproduction shifts towards light weights as harvesting increases, and its distribution changes from a single peak distribution under very low or high harvest rates, to a bimodal distribution under intermediate harvest rates. 3. These results suggest that, consistent with previous studies on aquatic species, harvesting-induced mortality may drive adaptive responses in ungulates by reducing the fitness benefits from adult survival and growth in favour of early and lightweight reproduction. 4. Selective harvesting for heavy animals has no additional effect on the evolutionarily stable strategy, suggesting that harvest rate is more important than the degree of selectivity in driving adaptive responses. However, selective harvesting of light females is positively associated with maturation weights even higher than those of a nonharvested population, probably due to the reduction in the fitness value of the offspring. 5. The average number of weight at maturation strategies in the population declines but the total number of strategies across all simulations increases with harvest rate, suggesting that harvesting-induced selection on weight at maturity overcomes the increase in strategy diversity expected from density-dependent release. 6. Yield initially increases with harvesting due to enhanced productivity of light females experiencing density-dependent release. However, it crashes under intense harvesting resulting in a population skewed to light, young and, therefore, less reproductive animals.     

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.     

The effect of seasonal harvesting on stage-structured population models

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose an exploited single-species model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Since birth pulse populations are often characterized with a discrete time dynamical system determined by its Poincaré map, we explore the consequences of harvest timing to equilibrium population sizes under seasonal dependence and obtain threshold conditions for their stability, and show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. Moreover, our results imply that the population can sustain much higher harvest rates if the mature fish is removed as early in the season (after the birth pulse) as possible. Further, the effects of harvesting effort and harvest timing on the dynamical complexity are also investigated. Bifurcation diagrams are constructed with the birth rate (or harvesting effort or harvest timing) as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, non-unique dynamics (meaning that several attractors coexist) and attractor crisis. This suggests that birth pulse, in effect, provides a natural period or cyclicity that makes the dynamical behavior more complex.This work is supported by National Natural Science Foundation of China (10171106)     

Population Dynamics of the Dioecious Amazonian Palm Mauritia flexuosa: Simulation Analysis of Sustainable Harvesting

The dioecious, tropical palm Mauritia flexuosa has high ecological and economic value, but currently some wild populations are harvested excessively, which is likely to increase. In this study, we investigated the population dynamics of this important palm, the effects of harvesting, and suggested sustainable harvesting regimes. Data were collected from populations in the Ecuadorian Amazon that were assumed to be stable. We used a matrix population model to calculate the density-independent asymptotic population growth rate (λ= 1.046) and to evaluate harvesting scenarios. Elasticity analysis showed that survival (particularly in the second and fifth size class) contributes more to the population growth rate, than growth and fecundity. To simulate a stable population at carrying capacity, density dependence was incorporated and applied to the seedling survival and growth parameters in the transition matrix. Harvesting scenarios were simulated with the density-dependent population models to predict sustainable harvesting regimes for the dioecious palm. We simulated the removal of only female palms and showed how both sexes are affected with harvest intensities between 15 and 75 percent and harvest intervals of 1–15 yr. By assuming a minimum female threshold, we demonstrated a continuum of sustainable harvesting schedules for various intensities and frequencies for 100 yr of harvest. Furthermore, by setting the population model's λ= 1.00, we found that a harvest of 22.5 percent on a 20 yr frequency for the M. flexuosa population in Ecuador is consistent with a sustainable, viable population over time.     

An assessment of the maximum sustainable yield of ivory from African elephant populations.

A general, logistic population model is used to explore the dynamics of harvested elephant populations. The model includes two features peculiar to elephant populations and the harvesting of ivory. First, because of the shape of the growth curve of tusks with age, the conversion factor that relates the number of elephants killed to the ivory yield in weight is not constant, but a function of the population size. Second, tusks from animals that die from natural causes can be retrieved and included in the total yield of ivory. The implications of the relationship between tusk size and age of an animal on the maximum sustainable yield in terms of ivory tonnage and in terms of the number of tusks are explored. The nonequilibrium implications of the tusk growth curve on the population dynamics under different harvesting strategies are also investigated. Results indicate that the maximum sustainable yield is achieved at very low harvest rates with population levels close to the pristine equilibrium. When tusks from animals that die of natural causes are included in the harvest, the maximum yield may, depending on the mortality and recruitment parameters, occur when there is no direct harvest.     

Maximum sustainable yield of populations with continuous age-structure.

We address the problem of finding the harvesting policy that will maximize the yield and maintain a population in a steady state. The population is characterized by continuous age classes and therefore follows differential equations. Here, we assume that the equations are linear (no density dependence). Two possible constraints are considered: either recruitment or total population are fixed to a constant. Under these conditions, the optimal policy is to harvest the fraction theta of a younger age class ? and to harvest totally an older age class b. The optimal solution (theta, ?, b) can be calculated explicitly if the fecundity and mortality schedules are given. The solution is compared to the simpler strategy of harvesting all individuals beyond a single age class a. It is shown that the latter strategy can be much less profitable than harvesting two age classes because it cannot take account of the different values of individuals according to their age.     

Effects of size‐ and sex‐selective harvesting: An integral projection model approach

Harvesting is often size‐selective, and in species with sexual size dimorphism, it may also be sex‐selective. A powerful approach to investigate potential consequences of size‐ and/or sex‐selective harvesting is to simulate it in a demographic population model. We developed a population‐based integral projection model for a size‐ and sex‐structured species, the commonly exploited pike (Esox lucius). The model allows reproductive success to be proportional to body size and potentially limited by both sexes. We ran all harvest simulations with both lower size limits and slot limits, and to quantify the effects of selective harvesting, we calculated sex ratios and the long‐term population growth rate (λ). In addition, we quantified to what degree purely size‐selective harvesting was sex‐selective, and determined when λ shifted from being female to male limited under size‐ and sex‐selective harvesting. We found that purely size‐selective harvest can be sex‐selective, and that it depends on the harvest limits and the size distributions of the sexes. For the size‐ and sex‐selective harvest simulations, λ increased with harvest intensity up to a threshold as females limited reproduction. Beyond this threshold, males became the limiting sex, and λ decreased as more males were harvested. The peak in λ, and the corresponding sex ratio in harvest, varied with both the selectivity and the intensity of the harvest simulation. Our model represents a useful extension of size‐structured population models as it includes both sexes, relaxes the assumption of female dominance, and accounts for size‐dependent fecundity. The consequences of selective harvesting presented here are especially relevant for size‐ and sex‐structured exploited species, such as commercial fisheries. Thus, our model provides a useful contribution toward the development of more sustainable harvesting regimes.     

Optimal harvesting strategies for timber and non-timber forest products in tropical ecosystems

Harvesting wild plants for non-timber forest products (NTFPs) can be ecologically sustainable–without long-term consequences to the dynamics of targeted and associated species–but it may not be economically satisfying because it fails to provide enough revenues for local people over time. In several cases, the same species can be harvested for NTFP and also logged for timber. Three decades of studies on the sustainability of NTFP harvest for local people’s livelihood have failed to successfully integrate these socio-economic and ecological factors. We apply optimal control theory to investigate optimal strategies for the combinations of non-lethal (e.g., NTFP) and lethal (e.g., timber) harvest that minimize the cost of harvesting while maximizing the benefits (revenue) that accrue to harvesters and the conservation value of harvested ecosystems. Optimal harvesting strategies include starting with non-lethal NTFP harvest and postponing lethal timber harvesting to begin after a few years. We clearly demonstrate that slow growth species have lower optimal harvesting rates, objective functional values and profits than fast growth species. However, contrary to expectation, the effect of species lifespan on optimal harvesting rates was weak suggesting that life history is a better indicator of species resilience to harvest than lifespan. Overall, lethal or nonlethal harvest rates must be <40 % to ensure optimality. This optimal rate is lower than commonly reported sustainable harvest rates for non-timber forest products.     

固定周期脉冲微分方程到状态依赖脉冲的转化及应用

本文研究了一类二维状态依赖脉冲微分方程的阶1周期解存在性和轨道稳定性条件．然后,将一维固定周期脉冲的微分方程转化为二维状态依赖脉冲微分方程,研究其阶一周期解的存在性和稳定性．作为应用,我们研究了固定周期常数收获的Logistic方程的动力学性质,以及两个固定周期注射药物单室扩散模型的动力学性质．     

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.     

Consequences of sex‐selective harvesting and harvest refuges in experimental meta‐populations

Harvesting for food or sport is often non‐random with respect to demographic state, such as size or life stage. The population‐level consequences of such selective harvesting depend upon which states are harvested and how those states contribute to population dynamics. We focused on a form of selective harvesting that has not previously been investigated in an experimental context: sex‐selective harvesting, a common feature of exploited, dioecious populations. Using simple metapopulations (two patches connect by dispersal) of sexually dimorphic Bruchid beetles in the laboratory, we contrasted the effects of female‐selective, male‐selective, and non‐selective harvesting over six generation of population dynamics. We also tested the ability of a harvest refuge (one patch of the metapopulation free from harvesting) to mitigate the effects of harvesting, and whether refuge effects interacted with sex selectivity. Sex‐selective harvesting significantly perturbed operational sex ratios and harvest refuges dampened these perturbations. Metapopulations assigned to male‐selective and non‐selective treatments were able to fully compensate for harvesting, such that their dynamics did not differ from non‐harvested controls. Only female‐selective harvesting led to significant reductions in population size and this effect was completely offset by dispersal from a harvest refuge. A two‐sex model confirmed that population dynamics are more sensitive to female vs. male harvesting, but suggested that higher levels of male harvest than included in our experiment would cause population decline. We discuss the roles of density‐dependent competition and frequency‐dependent sexual processes in the population response to sex‐selective harvesting.     

Hunting Bambi—evaluating the basis for selective harvesting of juveniles

Human harvesting is often a major mortality factor and, hence, an important proximate factor driving the population dynamics of large mammals. Several selective harvesting regimes focus on removing animals with low reproductive value, such as “antlered” harvests in North America and juvenile harvesting in many European countries. Despite its widespread use and assumed impact, the scientific basis of juvenile harvesting is scattered in the literature and not empirically well-documented. We give the first overview of demographic, evolutionary and practical management arguments for selective harvesting of juveniles. Furthermore, we empirically test two demographic arguments based on harvest statistics of Red Deer (Cervus elaphus) in seven European countries. P₁: Harvesting juveniles has little influence on harvest growth compared with harvesting adult females due to the lower reproductive value of juveniles than adult females; P₂: Harvesting of juveniles dampens variance in harvest due to lower and more variable natural survival rates of juveniles compared with adults. We found that harvesting juveniles has little effect on harvest growth rate, while harvesting adult females has a significant negative effect (consistent with P₁), but that increasing the proportion of juveniles in the harvest did not decrease the variability in harvest between years (P₂ not supported). Based on our empirical findings and overview of arguments, we discuss how the merits of juvenile harvesting may vary over time as populations move from a low density to a very high density state.     

Effects of strength and timing of harvest on seasonal population models: stability switches and catastrophic shifts

Population abundance of many species is controlled by a combination of density-dependent processes during different periods of the annual cycle. In the context of population exploitation or conservation programs, sequential density dependence has the potential to dramatically change population responses to harvesting. Looking for a better understanding of the potential effects of harvesting on the dynamics of seasonal populations, we carry out a theoretical analysis of a discrete model for a semelparous population with an annual cycle involving three discrete density-dependent events: breeding, natural mortality, and harvesting. Our study reveals how the interplay between the model parameters determines the importance of harvest timing on stability and population abundance, especially when two nontrivial stable equilibria coexist. We address the possibility for compensatory mortality and report different forms of the hydra effect, including non-smooth ones due to catastrophic shifts. These drastic switches may include hysteresis, which has important implications for conservation goals. Regarding variability, we show that increasing the harvesting effort may either stabilize or destabilize the population, and these effects strongly depend on harvest timing and natural mortality rates. Our results also emphasize the importance of sampling populations after every discrete event occurs during one cycle. Indeed, though the dynamics are not affected by census timing, the model shows that changes in population abundance in response to changes in harvesting pressure are substantially different depending on when population is sampled. Thus, a manager would receive different (and sometimes contradictory) messages depending on census time, which could lead to managing mistakes.     

一类功能性反应模型在常数收获下的动态特性

具有功能性反应的捕食与被捕食模型具有非常复杂的动态性质．特别是在常数收获下,该模型呈现了各种各样、纷杂多变的动态特性。其中包括正平衡点及其稳定性的变化、各种分叉的产生以及周期解和极限环的出现．本文重点研究了常数收获项对一类功能性反应模型的动态性能的影响,得到了该收获模型存在稳定正平衡点、产生分叉以及在Hopf分叉附近产生周期解和极限环的若干充分条件．     

A theory of stochastic harvesting in stochastic environments

We investigate how model populations respond to stochastic harvesting in a stochastic environment. In particular, we show that the effects of variable harvesting on the variance in population density and yield depend critically on the autocorrelation of environmental noise and on whether the endogenous dynamics of the population display over- or undercompensation to density. These factors interact in complicated ways; harvesting shifts the slope of the renewal function, and the net effect of this shift will depend on the sign and magnitude of the other influences. For example, when environmental noise exhibits a positive autocorrelation, the relative importance of a variable harvest to the variance in density increases with overcompensation but decreases with undercompensation. For a fixed harvesting level, an increasing level of autocorrelation in environmental noise will decrease the relative variation in population density when overcompensation would otherwise occur. These and other intricate interactions have important ramifications for the interpretation of time series data when no prior knowledge of demographic or environmental details exists. These effects are important whenever the harvesting rate is sufficiently high or variable, conditions likely to occur in many systems, whether the harvesting is caused by commercial exploitation or by any other strong agent of density-independent mortality.     

Temporal quota corrections based on timing of harvest in a small game species

Theoretical models have shown that the effect of removing a given proportion of the population can be profoundly different if the harvest takes place late in the season compared to early. We explore the effect of these differences using theoretical models based on the concept of demographic value and empirical data on seasonal patterns of natural mortality risk in two contrasting populations of willow ptarmigan in Norway. Based on the theoretical models, we found that changes in the timing of harvest have a much stronger effect in populations with relatively low annual survival compared to populations characterized by longevity typical for species with slow life histories. Also, the timing of harvest is more influential in cases with constant mortality hazards compared to a situation with density-dependent natural mortality. Empirical data from two study populations of willow ptarmigan showed large deviations from the theoretical predictions of models with both constant and density-dependent mortality hazards. There were also large differences in both the temporal pattern and magnitude of annual survival between the two ptarmigan populations (54 vs 26% annual survival). Site differences illustrate the importance of knowledge of both the magnitude and temporal pattern of natural mortality hazard to be able to correctly predict the effect of changing the timing of harvest in a population. In the two ptarmigan populations, we show how harvest quotas can be adjusted in accordance to the empirical estimates of natural mortality risk and how this determines the effects of shifting from harvesting early to late in the annual cycle.     

Harvesting,census timing and “hidden” hydra effects

Population control in some form of harvesting might be expected to reduce population size, but quite the opposite can happen due to the hydra effect. This phenomenon describes an increase in population size with increased mortality. One mechanism causing hydra effects is the temporal separation of (i) harvesting and (ii) density-dependent reproduction. Here we consider discrete-time models of these two processes. It is commonly believed that harvesting needs to precede reproduction for a hydra effect to occur. We show that, by contrast, hydra effects also take place for harvest after reproduction. Due to the timing of population census, however, the hydra effect will not be measured and thus remains ‘hidden’. As a consequence, managers may miss out on the opportunity to increase both the yield and the remaining stock of renewable resources. If harvesting aims at controlling pest species, management interventions may backfire in the sense that the pest increases rather than decreases—and, to make things even worse, this may actually go unnoticed. To remedy these undesirable consequences, we propose a modelling framework that can reveal hidden hydra effects. Our results are based on rigorous mathematical proofs that the order of two events does not matter for standard harvesting/hunting strategies.