Optimal Harvesting Policy Research Articles

Dynamics Analysis and Optimality in Selective Harvesting Predator-Prey Model With Modified Leslie-Gower and Holling-Type II

Abstract In this work, we consider the optimal harvesting and stability problems of a prey-predator model with modified Leslie-Gower and Holling-type II functional response. The model is governed by a system of three differential equations which describe the interactions between prey, predator and harvesting effort. Boundedness and existence of solutions for this system are showed. The existence and local stability of the possible steady states are analyzed and the conditions of global stability of the interior equilibrium are established by using the Lyapunov function, we prove also the occurrence of Hopf bifurcation at this point. By using the Pontryagin’s maximal principle, we formulate and we solve the problem of the optimal harvest policy. In the end, some numerical simulations are given to support our theoretical results.

Optimal harvesting policy of an inland fishery resource under incomplete information

AbstractA new mathematical model for finding the optimal harvesting policy of an inland fishery resource under incomplete information is proposed in this paper. The model is based on a stochastic control formalism in a regime‐switching environment. The incompleteness of information is due to uncertainties involved in the body growth rate of the fishery resource: a key biological parameter. Finding the most cost‐effective harvesting policy of the fishery resource ultimately reduces to solving a terminal and boundary value problem of a Hamilton‐Jacobi‐Bellman equation: a nonlinear and degenerate parabolic partial differential equation. A simple finite difference scheme for solving the equation is then presented, which turns out to be convergent and generates numerical solutions that comply with certain theoretical upper and lower bounds. The model is finally applied to the management of Plecoglossus altivelis, a major inland fishery resource in Japan. The regime switching in this case is due to the temporal dynamics of benthic algae, the main food of the fish. Model parameter values are identified from field measurement results in 2017. Our computational results clearly show the dependence of the optimal harvesting policy on the river environmental and biological conditions. The proposed model would serve as a mathematical tool for fishery resource management under uncertainties.

Theoretical Analysis of an Imprecise Prey-Predator Model with Harvesting and Optimal Control

In our present paper, we formulate and study a prey-predator system with imprecise values for the parameters. We also consider harvesting for both the prey and predator species. Then we describe the complex dynamics of the proposed model system including positivity and uniform boundedness of the system, and existence and stability criteria of various equilibrium points. Also the existence of bionomic equilibrium and optimal harvesting policy are thoroughly investigated. Some numerical simulations have been presented in support of theoretical works. Further the requirement of considering imprecise values for the set of model parameters is also highlighted.

The impact of provision of additional food to predator in predator–prey model with combined harvesting in the presence of toxicity

In this article, the global dynamics of a predator–prey system incorporating combined harvesting has been investigated. The additional food is being provided to the predator, accordingly, modified Holling type-II functional response is considered in the model. The presence of toxicants affect the quality of food for both the species, reducing their growth. The steady states of the system are obtained under some suitable conditions. The local and global dynamics are explored. The conditions for permanence and existence for bionomic equilibrium of the system have been investigated. It is also observed that the system exhibits local bifurcations i.e., transcritical, Hopf, saddle-node as well as global bifurcations i.e., Bogdanov–Takens bifurcation and generalized Hopf bifurcation with respect to the suitable set of parameters. Optimal harvesting policy is discussed with the help of Pontryagin’s maximum principle to preserve both the species from extinction and maintain a sustainable fishery. Numerical simulations are carried out for the suitable choice of parameters.

Simulation of Optimal Harvesting of Three Species Ecological Model with Closed Interval of Biological Parameter Using by Liapunov Function

The paper presents the study of three species ecological model with Prey N 1 , predator N 2 and competitor to the Predator N 3 and neutral with the predator N 2 with imprecise biological parameters. The model is characterized by a set of first order nonlinear ordinary differential equations. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. Equilibrium points of the model are identified, the local stability is discussed using Routh - Hurwitz criteria and global stability by Liapunov function. The existence of bionomic equilibrium of the system has been discussed and optimal harvesting policy is given using Pontryagin’s maximum principle. The stability analysis is supported by Numerical simulation using Mat lab. Keywords: Prey; Predator; Competitor to the predator; Equilibrium points; interval number, Stability of the equilibrium points; Bionomic Equilibrium; Optimal harvesting policy; Pontryagin’s maximum principle; Numerical simulation using mat lab. DOI : 10.7176/IEL/9-1-04

Simulation of Optimal Harvesting of Three Species Ecological Model with Closed Interval of Biological Parameter using by using First Order Nonlinear Ordinary Differential Equations

The paper presents the study of three species ecological model with Prey N 1 , predator N 2 and competitor to the Predator N 3 and neutral with the predator N 2 with imprecise biological parameters. The model is characterized by a set of first order nonlinear ordinary differential equations. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. Equilibrium points of the model are identified, the local stability is discussed using Routh - Hurwitz criteria and global stability by Liapunov function. The existence of bionomic equilibrium of the system has been discussed and optimal harvesting policy is given using Pontryagin’s maximum principle. The stability analysis is supported by Numerical simulation using Mat lab. Keywords : Prey; Predator; Competitor to the predator; Equilibrium points; interval number, Stability of the equilibrium points; Bionomic Equilibrium; Optimal harvesting policy; Pontryagin’s maximum principle; Numerical simulation using mat lab. DOI : 10.7176/JAAS/52-02

Dynamical analysis and optimal harvesting of a stochastic three-species cooperative system with delays and L\xe9vy jumps

A three-species cooperative system with time delays and Lévy jumps is proposed in this paper. Firstly, by comparison method and inequality techniques, we discuss the stability in mean and extinction of species, and the stochastic permanence of this system. Secondly, by applying asymptotic method, we investigate the stability in distribution of solutions. Thirdly, utilizing ergodic method, we obtain the optimal harvesting policy of this system. Finally, some numerical examples are given to illustrate our main results.

Dynamics of a Harvested Prey–Predator Model with Prey Refuge Dependent on Both Species

The present paper deals with a prey–predator model with prey refuge in proportion to both species, and the independent harvesting of each species. Our study shows that using refuge as control, it can break the limit cycle of the system and reach the required state of equilibrium level. We have established the optimal harvesting policy. The boundedness, feasibility of interior equilibria and bionomic equilibrium have been determined. The main observation is that the coefficient of refuge plays an important role in regulating the dynamics of the present system. Moreover, the variation of the coefficient of refuge changes the system from stable to unstable and vice-versa. Some numerical illustrations are given in order to support our analytical and theoretical findings.

Modeling and analysis of inhibitory effect in plankton–fish model: Application to the hypertrophic Swarzedzkie Lake in Western Poland

The hypertrophic Swarzedzkie Lake, Poland, is characterized by high species diversity, abundance and biomass of both phytoplankton and zooplankton. A relationship has been observed only with size groups and/or living forms of a few taxonomical groups of phytoplankton and a negative effect is found in zooplankton growth rate due to these groups of phytoplankton. This paper is devoted to mathematical study of such influences of toxic grouped phytoplankton on zooplankton and fish populations. The tri-trophic reaction–diffusion model incorporates Holling type III and Monod–Haldane functional response for three state variables namely, phytoplankton, zooplankton and fish. In order to investigate the dynamics of model, we have analyzed local and global stability of both non-spatial and spatial model systems. We have proved Hopf-bifurcation for non-spatial model and discussed maximum sustainable yield as well as optimal harvesting policy for maximizing economic gain. Different conditions for Turing instability have been obtained for spatial model. Numerical simulation is carried out for both temporal and spatial models. The temporal model is numerically analyzed with the help of phase portraits, time evaluation and bifurcation diagrams. Very beautiful Turing patterns are observed for the diffusion induced model. Influences of inhibitory effect are numerically investigated in both zooplankton and fish populations. This mathematical study suggests that inhibitory effect is able to destabilize homogeneous steady state and also able to produce chaos in plankton–fish system.

Optimal harvesting policy of a prey–predator model with Crowley–Martin-type functional response and stage structure in the predator

In this paper, a three-dimensional dynamical model consisting of a prey, a mature predator, and an immature predator is proposed and analysed. The interaction between prey and mature predator is assumed to be of the Crowley–Martin type, and both the prey and mature predator are harvested according to catch-per-unit-effort (CPUE) hypothesis. Steady state of the system is obtained, stability analysis (local and global both) are discussed to explore the long-time behaviour of the system. The optimal harvesting policy is also discussed with the help of Pontryagin's maximum principle. The harvesting effort is taken as an effective control instrument to preserve prey and predator and to maintain them at an optimal level.

Stochastic predator-prey model with Leslie-Gower and Holling-type II schemes with regime switching

A predator-prey model with Leslie-Gower and Holling-type II schemes with regime switching will be considered, that is, both white and color noises are taken into account. We firstly show that there exists a globally unique solution to the stochastic predator-prey model by use of the comparison theorem. Then, asymptotic properties of the system will be examined and the conditions under which the system is stochastically persistent will be given. Moreover, lastly, we analyze the optimal harvesting policy of the stochastic prey-predator model with Markovian switching.

About the optimal harvesting of a fuzzy predator–prey system: a bioeconomic model incorporating prey refuge and predator mutual interference

To understand roles of fuzzy biological parameters, in this paper, we propose a fuzzy predator–prey harvesting model that incorporates both the effects of prey refuge and predator mutual interference. Using the triangular fuzzy numbers for the imprecise parameters, we first study the existence of the feasible equilibria and their global stability and then discuss the bionomic equilibrium and the possible optimal harvesting policy. Numerical simulations are carried out to illustrate our analytical results, based on which discussions and conclusions are made. They show that the fuzziness of biological parameters can greatly affect the dynamics of the ecological system.

Dynamics of a stochastic delay competitive model with harvesting and Markovian switching

This work is concerned with a two-species stochastic state-switching competitive population model with distributed delays and harvesting. First, necessary and sufficient criteria for the existence of a unique ergodic stationary distribution of the system are established. Then necessary and sufficient criteria for the existence of the optimal harvesting policy are given, and the explicit expression of the optimal harvesting policy is obtained. Finally, some effects of the state-switching on the persistence, extinction and optimal harvesting strategy of the system are discussed with the help of several simulations.

A multicriteria stochastic optimization framework for sustainable forest decision making under uncertainty

A core process in forestry planning corresponds to the design of optimal harvesting policies along with road network layouts. In the most common setting, decision makers seek for solutions that maximize the profit of the forest while respecting operative and market constraints. Due to the long-term nature of the industry, the inherent uncertainty in both forest growth and market conditions should be taken into account. Nowadays, forest planning must target towards a sustainable management; the maximization of carbon sequestration and the minimization of land erosion are two common environmental goals.The planning challenge addressed in this paper integrates uncertainty of future forest growth and timber prices with the need for considering three criteria; net-present value, carbon sequestration, and land erosion caused by the road construction within the forest. By using mathematical programming tools and stochastic optimization techniques, we develop a stochastic multicriteria model that enables decision makers to have not only one, but a pool of long term planning policies.Moreover, a risk-averse variant of the framework is also considered. To the best of our knowledge, this is the first time that this type of forestry planning setting, which responds to the new challenges of the industry, is addressed.The proposed tool is used on an eucalyptus forest located in Portugal; the obtained results show the benefit of the proposed framework for producing a pool of sustainable forest plans with efficient trade-offs among the three considered criteria.

Dynamics of a predator–prey model with harvesting and reserve area for prey in the presence of competition and toxicity

The aim of this paper is to present and study a three-dimensional continuous time dynamical system modeling a predator–prey with harvesting and reserve zone for the prey in the presence of competition and toxicity. We first prove that our model is ecologically and mathematically well-posed. In addition, the stability analysis is investigated by direct and indirect Lyapunov methods. By using the Pontryagin’s maximum principle, an optimal harvesting policy is established. Furthermore, numerical simulations are given in order to illustrate our theoretical results.

English

English

Harvesting the Commons

We study a socio-ecological model in which a continuum of consumers harvest a common property renewable natural resource. Markov perfect Nash equilibria of the corresponding non-cooperative game are derived and are compared with collectively optimal harvesting policies. The underlying mechanisms that drive open-access commons in our model are shaped by population size, harvesting costs, and the ecosystem’s productivity. If other things equal population is small relative to harvesting costs, unmanaged commons do not face destruction. More strikingly, they are harvested at the collectively optimal rate. Property rights do not matter in that parametric regime because the resource has no social scarcity value. However, if other things equal population is large relative to harvesting costs, open-access renewable natural resources suffer from the tragedy of the commons. Property rights matter there because the resource has a social scarcity price. The population size relative to harvesting costs at which the socio-ecological system bifurcates is an increasing function of the ecosystem’s productivity. A sudden crash in productivity, population overshoot, or decline in harvesting costs can tip an unmanaged common into ruin. The model provides a way to interpret historical and archaeological findings on the collapse of those societies that have been studied by scholars.

Analysis of a Fishery Model with two competing prey species in the presence of a predator species for Optimal Harvesting

A harvesting fishery model is proposed to analyze the effects of the presence of red devil fish population, as a predator in an ecosystem. In this paper, we consider an ecological model of three species by taking into account two competing species and presence of a predator (red devil), the third species, which incorporates the harvesting efforts of each fish species. The stability of the dynamical system is discussed and the existence of biological and bionomic equilibrium is examined. The optimal harvest policy is studied and the solution is derived in the equilibrium case applying Pontryagin’s maximal principle. The simulation results is presented to simulate the dynamical behavior of the model and show that the optimal equilibrium solution is globally asymptotically stable. The results show that the optimal harvesting effort is obtained regarding to bionomic and biological equilibrium.

Dynamics analysis of a predator–prey system with harvesting prey and disease in prey species

ABSTRACTIn this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.

Numerical Computation of Robust Optimal Harvesting Policy in a Fish Pond under Uncertainty

Numerical Computation of Robust Optimal Harvesting Policy in a Fish Pond under Uncertainty