Constant Rate Harvesting Research Articles

Dynamics in a ratio-dependent predator–prey model with predator harvesting

The objective of this paper is to study systematically the dynamical properties of a ratio-dependent predator–prey model with nonzero constant rate predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov–Takens bifurcation), the subcritical and supercritical Hopf bifurcations. These results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in [D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737–753]. Biologically, it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction, and remove the singularity of the origin, which was regarded as “pathological behavior” for a ratio-dependent predator prey model in [P. Yodzis, Predator–prey theory and management of multispecies fisheries, Ecological Applications 4 (2004) 51–58].

Stability and bifurcation in a harvested one-predator–two-prey model with delays

Abstract It is known that one-predator–two-prey system with constant rate harvesting can exhibit very rich dynamics. If such a system contains time delayed component, it can have more interesting behavior. In this paper we study the effects of the time delay on the dynamics of the harvested one-predator–two-prey model. It is shown that time delay can cause a stable equilibrium to become unstable. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhaes. An example is given and numerical simulations are finally performed for justifying the theoretical results.

Bifurcations of a Ratio-Dependent Predator-Prey System with Constant Rate Harvesting

The ratio-dependent predator-prey model exhibits rich interesting dynamics due to the singularity of the origin. The objective of this paper is to study the dynamical properties of the ratio-dependent predator-prey model with nonzero constant rate harvesting. For this model, the origin is not an equilibrium. It is shown that numerous kinds of bifurcation occur for the model, such as the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov--Takens bifurcation, the homoclinic bifurcation, and the heteroclinic bifurcation, as the values of parameters of the model vary. Hence, there are different parameter values for which the model has a limit cycle, or a homoclinic loop, or a heteroclinic orbit, or a separatrix connecting a saddle and a saddle-node. These results reveal far richer dynamics compared to the model with no harvesting.

Numerical investigation of the optimal control rule for decision-making in fisheries management

: Recently, control rules (CR) have been widely used for decision-making of fisheries management. A CR describes fishing pressure (usually fishing mortality, F) as a function of some other variable(s) related to the status of the stock. The prevailing CR aim to achieve FMSY. The main idea of CR is to change fishing pressure based on stock conditions. Therefore, it seems irrational that the goals of prevailing CR are defined by the constant harvesting rate (CHR) strategy. The objective of the present study was to develop a method by which to numerically search for the optimal CR, apart from the traditional biological reference points based on CHR. I also compared the performance of the optimal CR and those of the prevailing CR (FMSY, 75%FMSY, New Management Procedure of the International Whaling Committee and the CR in US guidelines). The result suggested that the limitation of maximum fishing mortality was effective for stocks with low natural mortality and large stock size estimation error, while the biomass threshold was effective for stocks with short longevity and small estimation error. This means that the optimal CR is sensitive to the life history and uncertainty of the stock. Therefore, there seems to be no ‘one-size-fits-all’ rule for fish stocks.

Predator-prey models with delay and prey harvesting.

It is known that predator-prey systems with constant rate harvesting exhibit very rich dynamics. On the other hand, incorporating time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of the harvesting rate and the time delay on the dynamics of the generalized Gause-type predator-prey models and the Wangersky-Cunningham model. It is shown that in these models the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities, while the harvesting rate has a stabilizing effect on the equilibrium if it is under the critical harvesting level. In particular, one of these models loses stability when the delay varies and then regains its stability when the harvesting rate is increased. Computer simulations are carried to explain the mathematical conclusions.

Stabilization of a Chaotic Predator-Prey System Considering Fractal Boundaries

In this paper, the control scheme, which stabilizes the oscillating behavior including chaos, is discussed associated with a discrete predator-prey system described by the two-dimensional nonlinear difference equation. First, the mathematical model is given based on the stock-recruitment concept. It is shown through the stability analysis that the constant and constant rate harvestings of predators contribute to prevent oscillating behavior of the system. Secondly, recognizing the fact that a set of initial conditions of the system often exhibits the fractal boundary, its mechanism is clarified and the necessary condition for existence of fractal boundaries is analytically obtained. Finally, it is shown that, as the amount of the constant harvesting of predators increases, fractal boundaries disappear, while, under the constant rate harvesting, the complexity of fractal boundaries increases.

PERSISTENCE IN DISCRETE MODELS OF A POPULATION WHICH MAY BE SUBJECTED TO HARVESTING

We consider a class of one‐dimensional maps as discrete models of a single population. A general criterion for persistence of a population is obtained, and this criterion is shown to be valid for many models proposed in the literature. These results are extended to models which are periodic and to models which include constant rate harvesting.

比例制御入力を受ける離散型捕食-被食者系のカオス的挙動

By motivating the fact that discrete predator-prey systems often exhibit chaotic behavior, in this paper, control problems of a class of discrete predator-prey systems are studied.By examining the stability of an equilibrium associated with the predator-prey system, it is first shown that the harvesting or supplying of predators, whose amount is proportional to the number of predator, is effective for preventing oscillations of populations. It is also shown that if the growth rate of predators is too large and the population of predators at the equilibrium is too large, then populations oscillate and these oscillations are stabilized by the harvesting of predators. On the other hand, if the growth rate of predators is too small, the supplying of predators is effective. The optimal harvesting policy yielding the maximum average harvest of predators is also derived.Furthermore, by introducing the concept of the invariant domain, the constant rate harvesting investigated here is compared with the constant harvesting, and the former is shown to have an advantage from viewpoints of the conservation of ecosystems.The validity of theoretical results presented here is demonstrated by numerical experiments with various values of parameters.

Constant-Rate Harvesting of Age-Structured Populations

We consider the nonlinear age-dependent population growth model introduced by Gurtin and MacCamy [Arch. Rat. Mech. Anal., 54 (1974), pp. 281–300] with birth and death moduli depending on age and total population size, to which is added a harvest of members at a rate which is constant in time but may depend on the age of members being harvested. We formulate the problem as a pair of equations for the total population size and the birth rate, and discuss the behaviour of solutions when the birth and death moduli depend on either the age or the total population size, but not both.

Parametric optimization with dynamic systems

Parametric optimization with dynamic systems may be viewed using two perspectives. If it is viewed as a special case within optimal-control theory, then one set of necessary conditions is obtained. If it is viewed as an extension of classical parameter- optimization theory, then a different set of necessary conditions are obtained. We show here that the two sets of necessary conditions are, in fact, equivalent. However, numerical utilization of the conditions differ significantly with respect to difficulty in obtaining a solution. The necessary conditions obtained from parameter-optimization theory are generally easier to implement. This latter observation is demonstrated with the aid of two examples. One example involves environmental adaptation of annual plants, and the other involves constant-rate harvesting of a prey-predator system.

Coexistence properties of some predator-prey systems under constant rate harvesting and stocking

The global behaviour of a class of predator-prey systems, modelled by a pair of non-linear ordinary differential equations, under constant rate harvesting and/or stocking of both species, is presented. Theoretically possible structures and transitions are developed and validated by computer simulations. The results are presented as transition loci in the F-G (prey harvest rate-predator harvest rate) plane.

Boundedness of solutions of predator-prey systems

Under minimal assumptions, we establish boundedness of every solution of a predator-prey system with constant rate harvesting or stocking of either or both species. This leads to an extension of the classical Kolmogorov theorem on asymptotic behavior of solutions of predator-prey systems.

Forest management policies and elk summer carrying capacity in theAbies amabilis forest, western Washington

Effects of different forest management policies on long-term vegetation development are incorporated into a computer simulation of a western Washington watershed. The response of a Rocky Mountain elk(Cervus canadensis nelsoni) population to these vegetation changes is simulated using information on their differential use of vegetation types. Simulations include: 1) a cessation of timber harvesting leading to an immediate decline in elk members resulting from a reduction in summer habitats; 2) a stable elk population, similar to the present one, resulting from a constant timber harvesting rate, and 3) initial increases in herd size, followed within 50 years by a sharp decrease, resulting from cutting all old growth timber within the next decade and then a cessation of timber harvestings.

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→∞

Stability of some population models with delay

The stability of equilibrium of some single-species and predator-prey population models with delay and constant-rate harvesting is studied. It is suggested that the qualitative behavior of the one- and the two-dimensional model may be quite different from each other, and also that the two models may react quite differently to harvesting.

Constant rate harvesting of populations governed by Volterra integral equations

1. The effect of removing members from a population at a constant time rate has been studied for population sizes which are modelled by ordinary differential equations [2]. Such models represent populations for which it is assumed that the time rate of growth of population depends only on the population size. A more realistic model also assumes a probability of death depending on age which leads to a Volterra integral equation. Such models have been considered for populations, as well as for the number of members of a given population who are afflicted with a disease, from which recovery confers no immunity against reinfection [ 1, 43. In this paper we examine the effect of constant rate harvesting on such a model, and show that qualitatively the effect is similar to the effect of harvesting on an ordinary differential equation model. That is, there is a critical harvest rate which leads to extinction of the species, and for harvest rates less than the critical one it is possible to calculate the limiting population. The new feature of the more refined model considered here is that these calculations can be made directly in terms of observable parameters, rather than on such quantities as maximum unharvested growth rates which were used for calculation from the ordinary differential equation model in [2], and which are derived from computer simulations rather than directly from experiment.

Constant rate population harvesting: Equilibrium and stability

The paper studies the effect of constant rate harvesting on population models described by ordinary differential equations. The dependence of limits of populations on the harvest rate, the possibility of extinction in finite time, and the effect of the introduction of a time-lag delay on stability are studied. Various models for one population are analysed, and a geometric technique for the study of two-population models is discussed.