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Persistence and optimal harvesting of prey-predator model with Holling Type III functional response
M Agarwal, R Pathak
International Journal of Engineering, Science and Technology , 2012,
Abstract: The purpose of this work is to study the effect of harvesting on dynamics of prey- predator model with Holling Type III functional response. Mathematical analysis of the model equations with regard to the boundedness of solutions, existence of equilibria and their stabilities in both local and global manner are carried out. A combined harvesting policy for prey and predator species is discussed by using Pontrayagin’s Maximum principle and the effect of tax on prey-predator system is also shown. Butler-Mc Gehee lemma is used to identify the conditions which influence the persistence of the system. Finally, some numerical simulations are given to verify the mathematical conclusions.
Optimal Harvesting for an Age-structured Predator-prey System

He Zerong,

系统科学与数学 , 2006,
Abstract: This work analyzes the optimal harvesting of an age-based predator-prey system. Existence and uniqueness of non-negative solutions to the system and the continuous dependence of solution on control variable are proved. Existence of the optimal policy is discussed, and the optimality conditions are derived by means of normal cone and Dubovitskii-Milyutin theorem.
Optimal Harvesting of a Size-structured Predator-prey Model

LIU Yan,HE Ze-Rong,

数学物理学报(A辑) , 2012,
Abstract: This work is concerned with an optimal harvesting problem for a predator-prey model, in which the prey population is described by a first order partial differential equation (PDE) in a density function and the predator by an ordinary di?erential equation in total size. The existence and uniqueness of solutions to the state system and the dual system is proven via fixed point theorem. Necessary optimality conditions of first order are established by use of tangent-normal cones and dual system technique. The existence of a unique optimal control pair is derived by means of Ekeland’s variational principle. The resulting conclusion extends some existing results involving age-dependent populations.
Dynamic Behaviors of a Harvesting Leslie-Gower Predator-Prey Model
Na Zhang,Fengde Chen,Qianqian Su,Ting Wu
Discrete Dynamics in Nature and Society , 2011, DOI: 10.1155/2011/473949
Abstract: A Leslie-Gower predator-prey model incorporating harvesting is studied. By constructing a suitable Lyapunov function, we show that the unique positive equilibrium of the system is globally stable, which means that suitable harvesting has no influence on the persistent property of the harvesting system. After that, detailed analysis about the influence of harvesting is carried out, and an interesting finding is that under some suitable restriction, harvesting has no influence on the final density of the prey species, while the density of predator species is strictly decreasing function of the harvesting efforts. For the practical significance, the economic profit is considered, sufficient conditions for the presence of bionomic equilibrium are given, and the optimal harvesting policy is obtained by using the Pontryagin's maximal principle. At last, an example is given to show that the optimal harvesting policy is realizable.
Bifurcation Analysis of a Delayed Predator-Prey Model with Holling Type III Functional Response and Predator Harvesting  [PDF]
Uttam Das,T. K. Kar
Journal of Nonlinear Dynamics , 2014, DOI: 10.1155/2014/543041
Abstract: This paper tries to highlight a delayed prey-predator model with Holling type III functional response and harvesting to predator species. In this context, we have discussed local stability of the equilibria, and the occurrence of Hopf bifurcation of the system is examined by considering the harvesting effort as bifurcation parameter along with the influences of harvesting effort of the system when time delay is zero. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by applying the normal form theory and the center manifold theorem. Lastly some numerical simulations are carried out to draw for the validity of the theoretical results. 1. Introduction and Model Description Differential equation models for interactions between species are one of the classical applications of mathematics to biology, dating back to the first half of this century. The development and use of analytical techniques and the growth of computer power have progressively improved our understanding of these types of models. The study of population dynamics with harvesting is a subject of mathematical bioeconomics, which in turn related to the optimal management of renewable resources, Clark [1]. Generally the concept of optimal resource management is based on the standard cost benefit criterion which maximizes present values of net economic revenues. This criterion is relevant to both private and public management decisions, although the specification of costs and benefits are not necessarily the same in both cases. Regulation of exploitation of biological resources has become a problem of major concern nowadays in view of the dwindling resource stocks and the deteriorating environment. Exploitation reduces the biomass of the concerned species, exhibits oscillation, and even causes extinction of some other species. Rosenweig-MacAurtho model experiences oscillation under selective effort of Kar and Ghosh [2]. Legovic et al. [3] have concluded that harvesting the prey species at maximum level causes the extinction of the predator species in traditional prey-predator system. Kar and Ghosh [4] and Ghosh and Kar [5] show that harvesting the prey species at maximum sustainable yield (MSY) level never causes the extinction of the predator species in both ratio-dependent and Holling-Tanner prey-predator systems. It is shown that harvesting the prey species at MSY level may or may not drive the predator population to extinction if intraspecific competition is present among the predator species, Kar and Ghosh [4]. More recently, Ghosh and Kar [6]
Optimal Control of a Threatened Wildebeest-Lion Prey-Predator System in the Serengeti Ecosystem  [PDF]
T. D. Sagamiko, N. Shaban, C. L. Nahonyo, O. D. Makinde
Open Journal of Ecology (OJE) , 2015, DOI: 10.4236/oje.2015.54010
Abstract: We develop a two-species prey-predator model in which prey is wildebeest and predator is lion. The threats to wildebeest are poaching and drought while to lion are retaliatory killing and drought. The system is found in the Serengeti ecosystem. Optimal control theory is applied to investigate optimal strategies for controlling the threats in the system where anti-poaching patrols are used for poaching, construction of strong bomas for retaliatory killing and construction of dams for drought control. The possible impact of using a combination of the three controls either one at a time or two at a time on the threats facing the system is also examined. We observe that the best result is achieved by using all controls at the same time, where a combined approach in tackling threats to yield optimal results is a good approach in the management of wildlife populations.
Optimal Harvest of a Stochastic Predator-Prey Model  [cached]
Lv Jingliang,Wang Ke
Advances in Difference Equations , 2011,
Abstract: We firstly show the permanence of hybrid prey-predator system. Then, when both white and color noises are taken into account, we examine the asymptotic properties of stochastic prey-predator model with Markovian switching. Finally, the optimal harvest policy of stochastic prey-predator model perturbed by white noise is considered.
A dynamic reaction model of a prey-predator system with stage-structure for predator
Tapan Kumar Kar,Saroj Kumar Chattopadhyay
Modern Applied Science , 2010, DOI: 10.5539/mas.v4n5p183
Abstract: In this paper, we have considered a prey-predator model with stage-structure for predator and selective harvesting of prey species. A regulatory agency controls exploitation by imposing a tax per unit biomass of the prey species. The existence of steady states and their stability are studied. The problem of optimal harvesting policy is solved by using Pontryagin's maximal principle. It is also shown that time delay may cause a stable equilibrium to become unstable. Finally, some numerical simulations are carried out.
Harvesting and Hopf Bifurcation in a prey-predator model with Holling Type IV Functional Response
Manju Agarwal,Rachana Pathak
International Journal of Mathematics and Soft Computing , 2012,
Abstract: This paper aims to study the effect of Harvesting on predator species with time-delay on a Holling type-IV prey-predator model. Harvesting has a strong impact on the dynamic evolution of a population. Two delays are considered in the model of this paper to describe the time that juveniles of prey and predator take to mature. Dynamics of the system is studied in terms of local and Hopf bifurcation analysis. Finally, numerical simulation is done to support the analytical findings.
On the Effect of Switching, Predation and Harvesting on Systems Consisting of One Predator and Two Prey Species Which Live in Different Habitats
B.S. Bhatt,D.R. Owen,R.P. Jaju
Journal of Mathematics Research , 2011, DOI: 10.5539/jmr.v3n3p12
Abstract: The present work deals with two prey species living in two habitats and a predator specie which attacks the most abundant prey specie. The harvesting of both prey species is taken into account in the analysis. Non-zero equilibrium states have been examined with regard to their stability and the conditions for stability have been obtained. Using as a bifurcation parameter the conversion rate of prey to predator, conditions for a bifurcation to occur are obtained. A Hopf bifurcation theorem has been derived. In the investigations we used six forms of predation and harvesting rates to analyze the theory, however, we display graphical results for only one particular case.
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