Proportional Harvesting Research Articles

A Simple Predator-Prey Population Model with Rich Dynamics

A non-smooth switched harvest on predators is introduced into a simple predator-prey model with logistical growth of the prey and a bilinear functional response. If the density of the predator is below a switched value, the harvesting rate is linear; otherwise, it is constant. The model links the well studied predator-prey model with constant harvesting to that with a proportional harvesting rate. It is shown that when the net reproductive number for the predator is greater than unity, the system is permanent and there may exist multiple positive equilibria due to the effects of the switched harvest, a saddle-node bifurcation, a limit cycle, and the coexistence of a stable equilibrium and a unstable circled inside limit cycle and a stable circled outside limit cycle. When the net reproductive number is less than unity, a backward bifurcation from a positive equilibrium occurs, which implies that the stable predator-extinct equilibrium may coexist with two coexistence equilibria. In this situation, reducing the net reproductive number to less than unity is not enough to enable the predator to go extinct. Numerical simulations are provided to illustrate the theoretical results. It seems that the model possesses new complex dynamics compared to the existing harvesting models.

Effects of fuzzy harvesting on logistic growth models for uncertain ecological parameter

This paper presents the effects of two well known harvesting models namely quota harvesting and proportional harvesting on the logistic growth model under fuzziness. In many real life situations it is not possible to have a single growth rate of a species throughout the specified time span. Species are enhanced at different growth rates during their life span that arises due to nature of the soil, whether condition, rain fall, food supply etc. Similarly the total harvesting quantity for quota harvesting model and the proportion of the harvesting quantity for proportional harvesting quantity depend on the nature (need) of the harvester. So these are imprecise in nature, here we assume that the growth rate, constant harvesting quantity and proportion of harvesting quantity are uncertain in reality by considering imprecise by triangular fuzzy number. We find the equilibrium points, bifurcation value and discuss the stability of the equilibrium points of the models under fuzziness with different perspectives. Numerical examples are given to illustrate the harvesting models under fuzziness. Finally graphical illustrations are given to study the effects of harvesting on the logistic growth model under fuzziness.

Modelling the harvest of an insect pathogen

Ensuring a sustainable yield is essential for continued survival of a natural resource, however over-exploitation can easily occur. Therefore, understanding how increasing the harvesting rate affects the yield is vital. Harvesting of infected hosts in a host–pathogen system, for example the fungal pathogen Cordyceps sinensis which is harvested for medicinal use, has not been explored mathematically in the literature. We present a generalized host–pathogen model in which the infected host is harvested. Two strategies are explored; proportional harvesting at a constant rate and in an open-closed setting (a repeating cycle of a period of harvest followed by a period where the resource is left to recover). We present yield-effort curves for both strategies and find that open-closed harvesting affects the traditional yield-effort curve, with the system able to support a greater range of harvesting rates. Furthermore, host–pathogen systems may exhibit more complex population dynamics than single equation/species models, depending on the eigenvalues of the linearised system. In the open-closed setting we find that if there are complex eigenvalues in the absence of harvesting although small changes in the length of open season have little impact on the maximum sustainable yield, it can dramatically change the harvesting rate needed to achieve this. For proportional harvesting in a constant setting our model shows that if there are real eigenvalues in the absence of harvesting, then resilience–harvest relationship agrees with accepted theory, where as yield initially increases so too does the return time (a measure of the long-term resilience). However, when there are complex eigenvalues we see, counter to intuition, that the return time initially decreases whilst still providing increased yield. We also study the transient (short-term) reactivity, which shows that in both cases harvesting can initially decrease the reactivity. These results show that harvesting can in some instances enhance the ability of host–pathogen systems to respond to perturbations in both the short- and long-term.

Global asymptotical stability of a [formula omitted] dimensional Leslie–Gower predator–prey model

By constructing suitable Lyapunov function, we obtain the global asymptotical stability of n+1 dimensional Leslie–Gower predator–prey models. It is shown that the prey refuge and the proportional harvesting have no influence on the global asymptotical stability of the system, but they can change the position of the unique interior equilibrium. Finally, mathematic analysis shows that if we choose and control the harvesting efforts e,ei appropriately, we can not only harvest the biological resources and achieve the economic interest, but also protect the biological resources.

Lyapunov functions for two-species cooperative systems

Lyapunov functions for two-species mutualism models subject to proportional harvesting are introduced. Global stability of the unique coexisting equilibrium is thereby established.

Impact of maximum sustainable yield on competitive community

A system of n competing logistical species of the Volterra type under proportional harvesting strategy is analyzed. In case of selective harvesting, when the effort is adjusted to each species, the optimum effort may result in the total maximum sustainable yield (TMSY1). When it exists, reaching TMSY1 does not affect the system stability character, but it does affect the state, and hence some populations may reach too small a value to persist in nature. If competition is strong, species with smaller biotic potential may be driven to extinction. In case the system is harvested with a common harvesting effort, such as in trawler fishery, the total maximum sustainable yield (TMSY2) is smaller than TMSY1, and all the species with lower or equal biotic potential to the optimum harvesting effort will be driven to extinction. In this case a call for implementation of the MSY is equivalent to a call for the extermination of some species and it runs directly against the Convention on Biological Diversity (CBD, 1992). Therefore, all legal documents advocating MSY in ecosystems starting with the Johannesburg Plan of Implementation (JPI, 2002) must be urgently retracted and replaced with adaptive management which will respect CBD.

Impact of maximum sustainable yield on mutualistic communities

We analyze effects of reaching the maximum sustainable yield (MSY) in a system of n cooperating logistic populations under proportional harvesting strategy. In case a selective harvesting effort adjusted to each species is implemented, the optimum effort is smaller than half the biotic potential of each species. This results in the total maximum sustainable yield (TMSY1) which does not affect the persistence of the system. In case the system is harvested with a common harvesting effort which is approximately the same for all cooperating species, like in a trawler fishing, the optimum harvesting effort leads to the TMSY2, where TMSY1≥TMSY2. In addition, species with lower biotic potential and carrying capacity may be driven to extinction, especially if these species weakly cooperate with the rest of the system. Hence, a call for implementation of TMSY2 may be equivalent to a call for extermination of some species, and it runs against the Convention on Biological Diversity (CBD, 1992). Since the documents advocating MSY, starting with the Johannesburg Implementation plan (IP, 2002) and including the EU Green Paper (EU, 2009), use TMSY2, the call for MSY must be urgently retracted and replaced with a call for adaptive management that respects CBD.

DYNAMIC ANALYSIS OF A PERIODICALLY FORCED HOLLING-TYPE II TWO-PREY ONE-PREDATOR SYSTEM WITH IMPULSIVE CONTROL STRATEGIES

In this paper, we establish a two-competitive-prey and one-predator Holling type II system by introducing a proportional periodic impulsive harvesting for all species and a constant periodic releasing, or immigrating, for the predator at different fixed time. We show the boundedness of the system and find conditions for the local and global stabilities of twoprey-free periodic solutions by using Floquet theory for the impulsive differential equation, small amplitude perturbation skills and comparison techniques. Also, we prove that the system is permanent under some conditions and give sufficient conditions under which one of the two preys is extinct and the remaining two species are permanent. In addition, we take account of the system with seasonality as a periodic forcing term in the intrinsic growth rate of prey population and then find conditions for the stability of the two-prey-free periodic solutions and for the permanence of this system. We discuss the complex dynamical aspects of these systems via bifurcation diagrams.

Distributed delay logistic equations with harvesting

In this paper, we present four models for distributed delay logistic equations with harvesting. We study the behavior of the solutions and the stability of the equilibria in each models. Further, we consider the existence of bifurcation for a distributed delay logistic equation with harvesting and a discounted past. We show that the long-term outcomes are affected by the discounting of the past history, and the nature of the harvesting strategy (constant or proportional harvesting are considered). Finally, a numerical simulation is performed to illustrate the obtained results.

Impact of demersal fishery and evidence of the Volterra principle to the extreme in the Adriatic Sea

Published comparison of data from two sets of cruises in the Adriatic Sea: the first during 1948–1949 and the second during 1996–1998, has shown a significant decrease of demersal fish stocks during 50 years of fishery. Furthermore, it has been shown previously that the stock of Chondrichthyes has decreased while the stock of Osteichthyes has increased relative to Chondrichthyes and, for several species, the increase is in absolute values too. By using Schaefer's model (logistic population under proportional harvesting), the overfished categories are identified. It is also shown that observed changes in Chondrichthyes and Osteichthyes are in agreement with the Volterra principle down to the extinction of predators. Furthermore, by using the Gause competition model under an increasing fishing effort it is shown that an initially dominant competitor may become a marginal competitor, in agreement with observations. Improvements in the management of fish stocks in the Adriatic are proposed. Given the fact that no one so far has determined the optimum harvesting effort of any species set within a marine ecosystem, improvements apply for other seas too.

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.

Harvesting Strategies in a Fish Stock Dominated by Low-frequency Variability: The Norwegian Spring-spawning Herring (Clupea harengus)

Heavy positively autocorrelated natural fluctuations in a fisheries stock level are problematic for fisheries management, and collapses in the stock dynamics are difficult to avoid. In this paper, we compare three different harvesting strategies (proportional harvesting, threshold harvesting, and proportional threshold harvesting) in an autocorrelated and heavily fluctuating fishery — the Norwegian spring-spawning herring (Clupea harengus) — in terms of risk of quasi-extinction, average annual yield, and coefficient of variation of the yield. Contrary to general expectations, we found that the three strategies produce comparable yields and risks of quasi-extinction. The only observable difference was slightly higher yield and variation in the proportional threshold strategy when the yield is optimized. Thus, it remains an open question as how to characterize the circumstances when it is particularly needful to apply threshold levels in harvest policies.

SUSTAINABLE HARVESTING STRATEGIES OF WILLOW PTARMIGAN IN A FLUCTUATING ENVIRONMENT

Stochasticities in population dynamics as well as uncertainties in parameters often make it difficult to obtain reliable predictions of future population fluctuations. Here we use a long-term data set to model the fluctuations of a Willow Ptarmigan (Lagopus lagopus) population in Central Sweden. We use this model to examine how different harvest strategies affect mean annual yield, and how these results are influenced by uncertainties in parameter estimates. Restricted proportional harvest (each hunter is allowed to shoot only a limited number of grouse per day) gave slightly higher mean annual yields than proportional threshold harvesting (harvesting only a fixed proportion of the difference between the estimated population size and the threshold when this difference is positive). However, variance in annual yield was reduced by restricted proportional harvesting because periods with low population size became shorter. Uncertainties in population parameters did not affect which strategy was optimal although those uncertainties strongly influenced the expected yield and the uncertainties in the hunting statistics. Inaccuracies in population projections are therefore important to estimate and to model when developing sustained harvest strategies for fluctuating populations.

Optimal harvest of age‐structured populations of moose Alces alces in a fluctuating environment

We analysed sex‐ and age‐specific harvesting strategies of moose using an age‐structured population model that includes density dependence as well as environmental stochasticity. In order to find the strategy that maximises the mean annual yield we simulated the process over a large number of years. The mean annual yield is a function of the three parameters (number of harvested individuals of calves, adult (≥ ½ years old) bulls and adult females) that are involved in the definitions of the strategies. We compare, by numerical maximisation of a function in several variables, two harvest strategies: proportional harvesting, i.e. removal of a certain proportion of individuals in a given age‐and sex‐class, and threshold harvesting, i.e. all individuals of a given sex‐ and age‐class are harvested when the size of this subpopulation exceeds a certain threshold. In general, proportional harvest gives a smaller mean annual yield than threshold harvesting. The variance in the annual yield is, however, larger for threshold than for proportional harvesting. These differences between the two harvest strategies increase when the annual survival of calves is low, when there is high environmental stochasticity and when there is strong density regulation operating on survival. For both harvest strategies, the optimal harvest strategy involves high harvest of calves and adult bulls, whereas adult females should hardly be harvested.

A White-Tailed Deer Harvesting Strategy

We describe a harvest strategy that integrates a population dynamics model, a rationale for choosing a harvest rate, a means of implementing harvesting, and a change-in-ratio (CIR) population estimation technique. The size of the harvest is determined by selecting a proportion of the postharvest (residual) population (#) chosen between about zero and the intrinsic rate of increase in a given environment (rm), with a maximum sustained yield (MSY) at about 0 = r,/2 based on logistic growth, or about i = 0.5 based on McCullough's (1979:118, 1984) studies of the George Reserve white-tailed deer (Odocoileus virginianus) herd. The advantages of proportional harvesting are that the population stabilizes for all proportions, the location of the population along the sustained yield (SY) curve is unambiguous, and proportions are naturally self-correcting because harvests are tied to population size. Taking 2 separate single-type (sex) harvests provides a means for achieving the harvest quota and also yields a CIR population estimate based on observations of the proportion of antlered deer in the population at 3 times. The 2-stage CIR technique is robust to unequal observability, a major weakness of the traditional CIR methods. Our harvest strategy is most applicable where harvests can be controlled closely, and hunting seasons are relatively short. Field experiments are required to establish the efficacy of our harvest paradigm. J. WILDL. MANAGE. 52(4):589-595 Several components are needed to provide a framework for managing harvests of white-tailed deer to attain harvest and population goals. Among these are an underlying model of population dynamics that incorporates removals due to hunting, a rationale for choosing a harvest level, a means of implementing and controlling the removals, and a procedure to evaluate the success of the harvest strategy. Our objective is to integrate these requirements into a management paradigm based on a stock-recruitment model of population responses to harvesting (McCullough 1979:123-127, 1984), on 2 separate single-type (sex) removals during the hunting season, and on a 2-stage CIR population estimation technique (Pollock et al. 1985). We present a management paradigm, based on logic and theory that incorporates the concept of proportional harvesting. The 2-stage CIR is used to estimate population size, but any suitable estimation technique could be applied. We caution that our paradigm requires testing under field conditions. Our analysis begins with a deterministic model and then expands to consider briefly stochastic variability. Our strategy is most appropriate for herds closely controlled by managers. To support our rationale, we draw upon published literature and our experiences studying the deer herd on Remington Farms, Chestertown, Maryland. O r paradigm differs from the strategy developed by Hayne and Gwynn (1977), which uses the proportion of females in the harvest as a management objective. They reasoned that if th adult male harvest is inherently limited because hunter effort declines as males become scarce, then the female harvest could also be limited at a lower level by relating it to the male harvest. Their approach does not estimate population size or incorporate a measure of productivity, yet it has apparently been misinterpreted by some deer managers to reflect recruitment or changes in herd size (Downing 1981). Furthermore, Downing (1981) showed tha , alone, the proportion of females in the kill reveals little about the status of the herd. Our paradigm differs from Hayne and Gwynn's (1977) in that ours incorporates an estimate of the population and a measure of productivity. We thank Remington Farms and especially E. C. Soutiere and E. H. Galbreath for their encouragement and support. D. A. Adams, G. Caughley, P. D. Doerr, E. C. Franklin, and D. R. McCullough reviewed earlier drafts and provided helpful comments. This is paper 11498 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh. IPresent address: Florida Game and Fresh Water Fish Commission, Route 7, P.O. Box 440, Lake City, FL 32055.

The Multicohort Fishery Under Uncertainty

The multicohort fishery subject to random environmental disturbances is examined within a market framework. The free access problem is considered and optimal selective and proportional harvesting policies are discussed.

Pulse-fishing and stochastic equilibrium in the multicohort fishery

Pulse-fishing harvesting policies are shown to be optimal for a competitive firm managing a multicohort fishery subject to environmental disturbances. Selective harvesting and non-selective proportional harvesting policies are examined. It is shown that pulse-fishing implies convergence to a unique, invariant probability distribution on cohort biomass levels and environmental disturbances that is independent of the initial state of the system.

Response of predator-prey systems to nutrient enrichment and proportional harvesting

A predator-prey System is modelled by a pair of ordinary differential equations, and the qualitative effects of prey nutrient enrichment and predator harvesting at a rate proportional to the predator population size are studied. Some theoretical analysis concerning the stability of equilibrium points and the existence of stable limit cycles are included. Three models are examined as examples, and for two of them computer simulations are included to illustrate the changes in qualitative behaviour under nutrient enrichment and increase of harvesting effort. The essential difference between this study and our previous work on constant-rate harvesting (Brauer et al. 1976) is that, here, extinction of predators in finite time is impossible although the predator population may tend to zero as l→∞

Electrons provide means of analyzing metal surfaces: Machinery, Vol. 53. No. 5.

Published comparison of data from two sets of cruises in the Adriatic Sea: the first during 1948–1949 and the second during 1996–1998, has shown a significant decrease of demersal fish stocks during 50 years of fishery. Furthermore, it has been shown previously that the stock of Chondrichthyes has decreased while the stock of Osteichthyes has increased relative to Chondrichthyes and, for several species, the increase is in absolute values too.By using Schaefer's model (logistic population under proportional harvesting), the overfished categories are identified. It is also shown that observed changes in Chondrichthyes and Osteichthyes are in agreement with the Volterra principle down to the extinction of predators. Furthermore, by using the Gause competition model under an increasing fishing effort it is shown that an initially dominant competitor may become a marginal competitor, in agreement with observations.Improvements in the management of fish stocks in the Adriatic are proposed. Given the fact that no one so far has determined the optimum harvesting effort of any species set within a marine ecosystem, improvements apply for other seas too.