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.     

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.     

Should hunting mortality mimic the patterns of natural mortality?

With growing concerns about the impact of selective harvesting on natural populations, researchers encourage managers to implement harvest regimes that avoid or minimize the potential for demographic and evolutionary side effects. A seemingly intuitive recommendation is to implement harvest regimes that mimic natural mortality patterns. Using stochastic simulations based on a model of risk as a logistic function of a normally distributed biological trait variable, we evaluate the validity of this recommendation when the objective is to minimize the altering effect of harvest on the immediate post-mortality distribution of the trait. We show that, in the absence of compensatory mortality, harvest mimicking natural mortality leads to amplification of the biasing effect expected after natural mortality, whereas an unbiased harvest does not alter the post-mortality trait distribution that would be expected in the absence of harvest. Although our approach focuses only on a subset of many possible objectives for harvest management, it illustrates that a single strategy, such as hunting mimicking natural mortality, may be insufficient to address the complexities of different management objectives with potentially conflicting solutions.     

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.     

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.     

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)     

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.     

Density-Dependent Harvest Modeling for the Eastern Wild Turkey

Abstract Many current wild turkey (Meleagris gallopavo) harvest models assume density-independent population dynamics. We developed an alternative model incorporating both nonlinear density-dependence and stochastic density-independent effects on wild turkey populations. We examined model sensitivity to parameter changes in 5% increments and determined mean spring and fall harvests and their variability in the short term (3 yr) and long term (10 yr) from proportional harvesting under these conditions. In the long term, population growth rates were most sensitive to poult:female ratios and the form of density dependence. The nonlinear density-dependent effect produced a population that maximized yield at 40% carrying capacity. The model indicated that a spring or fall proportional harvest could be maximized for fall harvest rates between 0% and 13% of the population, assuming a 15% spring male harvest and 5% spring illegal female kill. Combined spring and fall harvests could be maximized at a 9% fall harvest, under the same assumptions. Variability in population growth and harvest rates increased uncertainty in spring and fall harvests and the probability of overharvesting annual yield, with growth rate variation having the strongest effect. Model simulations suggested fall harvest rates should be conservative (≤9%) for most management strategies.     

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.     

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 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.     

Culling Versus Density Effects in Management of a Deer Population

Abstract: Wildlife managers often manipulate hunting regulations to control deer populations. However, few empirical studies have examined the level of hunting effort (hunter-days) required to limit population growth and demographic effects through harvesting of females. Moreover, the relative importance of density effects on population growth has not been quantified. We reconstructed a sika deer [Cervus nippon] population over a period of 12 years (1990–2001) using age- and sex-specific harvest data. Using cohort analysis, we analyzed population dynamics, focusing on 1) the relationship between hunting effort and hunting-induced mortality rate, 2) relative contributions of hunting mortality and recruitment of yearlings to annual changes in population growth rate, and 3) annual variation in recruitment rate. Population size increased until 1998 and declined thereafter. The population growth rate changed more in response to annual changes in recruitment rate than hunting mortality rate. Temporal variation in recruitment rate was not controlled by birth rate alone; direct density dependence, intensities of hunting mortality for fawns, and for females (≥2 yr of age), which accounted for the fawn survival rate, were required as factors to explain temporal variation. Density effects on the recruitment rate were not strong enough to regulate the population within the study period; high hunting mortality, with intensive female harvesting, was necessary to prevent population growth. Hunting effort was a good predictor of the hunting mortality rate, and female harvest had a negative effect on the recruitment rate through fawn survival. We suggest that >3,500 hunter-days and prioritization of female harvesting are required to prevent increases in this deer population.     

Interaction effects between weather and space use on harvesting effort and patterns in red deer

Most cervid populations in Europe and North America are managed through selective harvesting, often with age‐ and sex‐specific quotas, with a large influence on the population growth rate. Less well understood is how prevailing weather affects harvesting selectivity and off‐take indirectly through changes in individual animal and hunter behavior. The behavior and movement patterns of hunters and their prey are expected to be influenced by weather conditions. Furthermore, habitat characteristics like habitat openness are also known to affect movement patterns and harvesting vulnerability, but how much such processes affect harvest composition has not been quantified. We use harvest data from red deer (Cervus elaphus) to investigate how weather and habitat characteristics affect behavioral decisions of red deer and their hunters throughout the hunting season. More specifically, we look at how sex and age class, temperature, precipitation, moon phase, and day of week affect the probability of being harvested on farmland (open habitat), hunter effort, and the overall harvest numbers. Moon phase and day of week were the strongest predictors of hunter effort and harvest numbers, with higher effort during full moon and weekends, and higher numbers during full moon. In general, the effect of fall weather conditions and habitat characteristics on harvest effort and numbers varied through the season. Yearlings showed the highest variation in the probability of being harvested on farmland through the season, but there was no effect of sex. Our study is among the first to highlight that weather may affect harvesting patterns and off‐take indirectly through animal and hunter behavior, but the interaction effects of weather and space use on hunter behavior are complicated, and seem less important than hunter preference and quotas in determining hunter selection and harvest off‐take. The consideration of hunter behavior is therefore key when forming management rules for sustainable harvesting.     

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.     

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.     

Ecological balance in the native population dynamics may cause the paradox of pest control with harvesting

We analyze a time-discrete mathematical model of host-parasite population dynamics with harvesting, in which the host can be regarded as a pest. We harvest a portion of the host population at a moment in each parasitism season. The principal target of the harvesting is the host; however, the parasite population may also be affected and reduced by a portion. Our model involves the Beverton-Holt type density effect on the host population. We investigate the condition in which the harvesting of the host results in an eventual increase of its equilibrium population size, analytically proving that the paradoxical increase could occur even when the harvesting does not directly affect the parasite population at all. We show that the paradox of pest control could be caused essentially by the interspecific relationship and the intraspecific density effect.     

Bifurcations and chaos in a predator-prey system with the Allee effect

It is known from many theoretical studies that ecological chaos may have numerous significant impacts on the population and community dynamics. Therefore, identification of the factors potentially enhancing or suppressing chaos is a challenging problem. In this paper, we show that chaos can be enhanced by the Allee effect. More specifically, we show by means of computer simulations that in a time-continuous predator-prey system with the Allee effect the temporal population oscillations can become chaotic even when the spatial distribution of the species remains regular. By contrast, in a similar system without the Allee effect, regular species distribution corresponds to periodic/quasi-periodic oscillations. We investigate the routes to chaos and show that in the spatially regular predator-prey system with the Allee effect, chaos appears as a result of series of period-doubling bifurcations. We also show that this system exhibits period-locking behaviour: a small variation of parameters can lead to alternating regular and chaotic dynamics.     

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.     

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.     

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.