Optimal Harvesting Policy Research Articles

Dynamical Analysis of a Beddington–DeAngelis Interacting Species System with Prey Harvesting

A reaction–diffusion interacting species system with Beddington–DeAngelis functional response that has been proposed in the environment of mathematical ecology, which provides the rise to spatial pattern formation, is investigated and associated with the models of deterministic dynamics. The dynamical behaviour of a generalist predator–prey system with linear harvesting of each species and predator-dependent functional response is fully analyzed. Conditions of stability behaviour of the interior equilibrium point are established properly. Furthermore, we have recognized that the unique positive equilibrium point of the system is globally stable via appropriate Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system. Also, we establish the conditions for the existence of bifurcation phenomena including a saddle-node bifurcation and a Hopf bifurcation. Subsequently, complete analysis regarding the impact of harvesting is carried out, and an interesting decision is that under some appropriate constraints, harvesting has immense impact on the final size of the interacting species. In addition, in accordance with Turing’s ideas on morphogenesis , our analysis shows that harvesting effort in a reaction–diffusion predator–prey system plays a vital function for geological conservation of interacting species. Finally, we discuss sufficient conditions for the existence of bionomic equilibrium point and the optimal harvesting policy attained by using the Pontryagin maximal principle.

Dynamical study of quadrating harvesting of a predator–prey model with Monod–Haldane functional response

In this study, we have investigated local and global dynamics of a modified Leslie–Gower predator–prey model with Monod–Haldane functional response, where prey is subjected to quadratic harvesting. It is found that the solutions of the proposed system are positive and bounded uniformly. The feasible equilibrium points are also obtained for some suitable and predefined conditions. It is observed that the system exhibits at most three non-zero interior equilibrium points for different choices of parameters under certain conditions. The dynamics of all these feasible equilibrium points have been analysed using Routh–Hurwitz criterion. Local bifurcations analysis such as transcritical and saddle-node bifurcations have been investigated using Sotomayor’s theorem. To demonstrate the analytical results, numerical simulations using some suitable data set are carried out. Optimal harvesting policy has been obtained using Pontryagin’s Maximum Principle to show that the species can be preserved from extinction and a sustainable fishery can be achieved.

Chaos Detection and Optimal Control in a Cannibalistic Prey–Predator System with Harvesting

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin’s maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level.

Optimal harvesting of an abstract population model with interval biological parameters

We study the optimal harvesting policy for fishery in the marine protected and unreserved areas. In the literature, it is generally assumed that the fish population follows a concrete growth law. In contrast, we consider an abstract model with migration from the reserved area to the unreserved one. Then we examine and analyze the existence and stability of a nontrivial equilibrium point of the model. We also discuss the bionomic equilibrium. After that, we use the Pontryagin maximum principle to obtain the optimal harvest policy, where, instead of the well-known Hamiltonian function, we use the current Hamiltonian function to ease the calculation. Finally, we give some numerical examples to further illustrate our statements, where we also find that in practice the impreciseness of the parameters can influence the existence of the system positive equilibrium.

On a predator-prey system interaction under fluctuating water level with nonselective harvesting

Abstract A predator-prey model interaction under fluctuating water level with non-selective harvesting is proposed and studied in this paper. Sufficient conditions for the permanence of two populations and the extinction of predator population are provided. The non-negative equilibrium points are given, and their stability is studied by using the Jacobian matrix. By constructing a suitable Lyapunov function, sufficient conditions that ensure the global stability of the positive equilibrium are obtained. The bionomic equilibrium and the optimal harvesting policy are also presented. Numerical simulations are carried out to show the feasibility of the main results.

Stochastic differential equations harvesting policies: Allee effects, logistic‐like growth and profit optimization

AbstractIn this article, stochastic differential equations are used to model the dynamics of a harvested population in the presence of weak Allee effects. Two optimal harvesting policies are presented, one with variable effort based on optimal control theory, which is for practical reasons inapplicable in a random environment, and the other with constant effort and easily applicable. For a logistic‐like model with weak Allee effects, we show that the optimal policy based on constant effort implies, in a suitable range of effort values, the existence of a steady‐state stochastic equilibrium with a stationary density, obtained explicitly here, for the population size. With this new result, we compare the performance of both policies in terms of the profit obtained over a finite time horizon. Using realistic data from a harvested population and a logistic‐type growth model, we quantify the profit reduction when choosing the optimal policy based on constant effort instead of the optimal policy based on variable effort. We also study the influence of the Allee effects strength.

STABILITY AND BIFURCATION ANALYSIS OF A DISCRETE PREY–PREDATOR MODEL WITH SQUARE-ROOT FUNCTIONAL RESPONSE AND OPTIMAL HARVESTING

In this paper, we have considered a discrete prey–predator model with square-root functional response and optimal harvesting policy. This type of functional response is used to study the dynamics of the prey–predator model where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The considered population model has three fixed points; one is trivial, the second one is axial and the last one is an interior fixed point. The first two fixed points are always feasible but the last one depends on the parameter value. The interior fixed point experiences the flip and Neimark–Sacker bifurcations depending on the predator harvesting coefficient. Finally, an optimal harvesting policy has been introduced and the optimal value of the harvesting coefficient is determined.

Dynamical analysis of a fuzzy phytoplankton–zooplankton model with refuge, fishery protection and harvesting

In this paper, a singular phytoplankton–zooplankton model with fuzzy parameters, refuge, fishery protection and harvesting is studied by regarding the imprecise biological parameters as one form of triangular fuzzy number. By using the utility function method, the transformed differential equation is reduced to a differential equation. In absence of economic profit, the existence of positive equilibria, local stability and global stability for the interior equilibrium of the present differential system are discussed, the conditions of Hopf bifurcation occurring at the positive equilibrium are given, and the corresponding nonlinear feedback controller is designed to eliminate Hopf bifurcation. In presence of economic profit, the corresponding differential algebraic model is proposed and some conditions of the existence of the interior equilibrium are derived. By regarding economic profit as bifurcation parameter, singular induced bifurcation occurring is investigated, and a linear controller is designed to remove such bifurcation. Furthermore, Pontryagin’s maximum principle has been used to obtain optimal harvesting policy to maximize the benefit as well as conservation of the ecosystem. Finally, some numerical simulations are given to demonstrate our theoretical results after each part. Results show that imprecise parameters not only affect the interior equilibrium and the bionomic equilibrium of system, but also affect the critical value of bifurcation and branch range.

Global dynamics of a Holling Type-III two prey–one predator discrete model with optimal harvest strategy

This paper deals with a discrete-time two prey–one predator system with Holling Type-III functional response, along with inter-specific competition between the prey and intra-specific competition among the predators, with nonlinear harvesting of a prey population. The conditions for existence and local stability of the equilibrium points and the sufficient conditions for global asymptotic stability of the interior equilibrium point are derived. Furthermore, optimal harvesting policy is obtained by extension of Pontryagin’s maximum principle to discrete system. Meanwhile, some numerical simulations are provided to corroborate the analytical results.

Harvesting of a prey-predator model fishery in the presence of competition and toxicity with two effort function

In this paper, we propose a predator-prey model with harvesting and reserved area for prey with the presence of competition and toxicity with two effort functions. First, we prove the boundedness of the solutions. Then, the existence is studied, as well as the local and global stability of the equilibria. Lyapunov proved this last with certain conditions. The optimal harvesting policy is discussed using the Maximum Principle of Pantryagin. Finally, we ensure our results by numerical simulations.

Dynamical Behavior of Two Toxic Releasing Competing Species in Presence of Predator

The present paper deals with a problem of two competing fish species where each of the species release a substance toxic for the other species. It is considered that each of the species migrate due to toxic substance released by other species. Conditions for the existence and stability of all feasible equilibrium points are obtained. The dynamical behavior of the system is examined in the presence of migration. Lastly, the optimal harvesting policy is obtained by using the Pontryagin’s maximum principle. MATLAB is used to simulate the system using hypothetical set of parametric values.

Dynamics of a fishery system in a patchy environment with nonlinear harvesting

In this paper, a stock‐effort dynamical model with two fishing zones is discussed. The nonlinear harvesting function is assumed depending upon stock size as well as fishing effort. The migration of fish is considered between two zones. The harvesting vessels also move between zones to increase their revenue. The movements of fish and fishing vessels between zones are assumed to take place at a faster time scale as compared with processes involving growth and harvesting occurring at a slow time scale. The aggregated model is obtained for total fish stock and fishing effort. This aggregated (reduced) model is analyzed analytically as well as numerically. Biological and bionomic equilibria of the system are obtained, and criteria for local stability or instability of the system are derived. The impact of levels of taxation T on the fish population and on the revenue earned by the fishery is investigated. An optimal harvesting policy is also discussed using the Pontryagin's maximum principle. The aggregated model also exhibits Hopf and transcritical bifurcation with respect to the bifurcation parameter tax T. Numerical simulations are presented to illustrate the results.

Determining optimal policies for sugarcane harvesting in Thailand using bi-objective and quasi-Newton optimization methods

In Thailand, the harvesting season for sugarcane usually begins in November and ends the following May. At the beginning of each harvesting season, the Royal Thai government sets the price of two types of sugarcane, namely fresh and fired, based on sweetness (sugar content) and gross weight of sugarcane delivered to the sugar factories. The aim of the present research is to determine optimal harvesting policies for the two types of sugarcane in sugarcane producing regions of Thailand in order to maximize revenue and minimize harvesting cost. In this paper, a harvesting policy is defined as the amount of each type of sugarcane harvested and delivered to the sugar factories during each 15-day period of a harvesting season. Two optimization methods have been used to solve this optimization problem, namely the ε-constraints method and a quasi-Newton optimization method. In the ε-constraints method, the problem is considered as a bi-objective optimization problem with the main objective being to determine harvesting policies that maximize the total revenue subject to upper bounds on the harvesting cost. In the quasi-Newton method, the aim is to determine the harvesting policy which gives maximum profit to the farmers subject to constraints on the maximum amount that can be cut in a 15-day period. The methods are used to determine optimal harvesting policies for the north, central, east, and north-east regions of Thailand for harvesting seasons 2012/13, 2013/14, and 2014/15 based on the data obtained from the Ministry of Industry and the Ministry of Agriculture and Co-operatives of the Royal Thai government.

Impact of Harvesting on a Bioeconomic Predator-Prey Fishery Model Subject to Environmental Toxicant.

The present paper studies a predator-prey fishery model which incorporates the independent harvesting strategies and nonlinear impact of an anthropogenic toxicant. Both fish populations are harvested with different harvesting efforts, and the cases for the presence and non-presence of harvesting effort are discussed. The prey fish population is assumed to be infected by the toxicant directly which causes indirect infection to predator fish population through the feeding process. Each equilibrium of the proposed system is examined by analyzing the respective local stability properties. Dynamical behavior and bifurcations are studied with the assistance of threshold conditions influencing the persistence and extinction of both predator and prey. Bionomic equilibrium solutions for three possible cases are investigated with certain restrictions. Optimal harvesting policy is explored by utilizing the Pontryagin's Maximum Principle to optimize the profit while maintaining the sustainability of the marine ecosystem. Bifurcation analysis showed that the harvesting parameters are the key elements causing fishery extinction. Numerical simulations of bionomic and optimal equilibrium solutions showed that the presence of toxicant has a detrimental effect on the fish populations.

Harvesting in a toxicated intraguild predator–prey fishery model with variable carrying capacity

Latterly, variable carrying capacity in the predator–prey population models has been widely studied to provide more realistic understanding about population dynamics. In the present paper, we discuss the bio-economic harvesting of an intraguild fishery model where the logistic carrying capacities of both predator and prey fish populations are variable based on the shared biotic resource. Distinct from most fishery models, we incorporate the independent harvesting strategies on the fisheries with different harvesting efforts by assuming them to have different economic values. In our model, the prey fish population is assumed to be infected directly by an anthropogenic toxicant from the environment while the predator is infected indirectly when they feed on prey. We investigate the local stability of trivial equilibria with Routh–Hurwitz criterion and the global stability of non-trivial equilibrium by constructing a Lyapunov function. Bifurcation and numerical analyses are presented to show distinctive result that a Hopf bifurcation occurs with respect to harvesting parameter instead of resource enrichment parameter as in the literature. Bionomic equilibrium is explored with several possibilities and restrictions. The nontrivial bionomic equilibrium is found to depend critically on the resource density. Finally, the optimal harvesting policy of the three dimensional model is derived by utilizing Pontryagin Maximum Principle. From this study, harvesting parameter is a crucial factor for the stability of the intraguild system. The objective is to obtain the optimal sustainable yield that provides maximum monetary profit while conserving the marine ecosystem.

Optimal harvesting of a stochastic commensalism model with time delay

This article is concerned with a two-species stochastic delay commensalism model with harvesting. Firstly, sufficient and necessary criteria for extinction and persistence of each species are established. Then sufficient and necessary criteria for the existence of a unique optimal harvesting policy are obtained, and the explicit form of the optimal harvesting policy is given. Finally, some influences of stochasticity on extinction, persistence and optimal harvesting policy are illustrated with several simulations.

Optimal harvesting policy of logistic population model in a randomly fluctuating environment

In this paper, we investigate the optimal harvest policy of a stochastic logistic population model. The value of this study lies in two aspects: mathematically, we establish a stochastic threshold theorem to govern whether the population persists or not. In the case of population persistence, we prove the existence, uniqueness and global asymptotic stability of the invariant density of the Fokker–Planck equation associated with the SDE model, and we further show the relation between these two models. In addition, we give the unique optimal harvesting effort and the corresponding maximum of expectation of sustainable yield, which gives us the profile of the optimal harvesting policy of the SDE model. Ecologically, we find that big harvesting effort or big intensity of noise will lead the population to extinct risk almost surely. In addition, we find that under a fixed randomness strategy and proper harvesting, the maximum sustainable yield increases systematically as the harvesting effort increases, but overexploitation will reduce the level of maximize sustainable yield and eventually make the whole population extinct with probability one. Hence in order to obtain the optimal harvesting policy, we must decrease the harvesting effort and the intensity of noise. Furthermore, we find that our parameter perturbation method in this paper is more beneficial to the exploitation of renewable resources than the classical one given by Beddington and May (1977). The results show that different perturbation method can exhibit different stochastic population dynamics.

An Optimal Harvesting of A Discrete Time System with Ricker Growth

In this study, the dynamic behavior of a sporadic model (prey-predator) is investigated with Ricker function growth in prey species. We also found the system has four fixed points. We set the conditions that required to achieve local stability of all fixed points. The rate of harvest in the case of being a fixed quantity in the community and the existence of the bionomic equilibruim in the absence of predator are discussed then the system is extended to an optimal harvesting policy. The Pontryagin’s maximum principle is used to solve the optimality problem. Numerical simulations have been applied to enhance the results of mathematical analysis of the system.

Bifurcation and bio-economic analysis of a prey-generalist predator model with Holling type IV functional response and nonlinear age-selective prey harvesting

Abstract In this paper, we have studied the dynamical behaviour of a Leslie-Gower model with Holling type-IV functional response and nonlinear prey harvesting where target prey for harvesting are above a particular age which is called maturation delay. The model system has been studied using qualitative analysis, bifurcation theory, bionomic equilibrium, MSY and optimal harvesting policy. We explain that the interior equilibrium is delay dependent and locally asymptotically stable. The age-selective harvesting system under goes a Hopf-bifurcation with respect to the maturation delay. Conditions for the MSY and existence of bionomic equilibrium are analyzed and Pontryagin's maximum principle has been used to find the path of a singular control. Difficult analytical results are featured by numerical examples to better understand the system in a simple way.

Optimal Power Allocation for Wireless Sensor Powered by Dedicated RF Energy Source

This paper studies a wireless-powered sensor network, where a sensor harvests energy from a dedicated radio-frequency (RF) energy source and transmits information to an information sink using the harvested energy. Two working modes are considered; One is the frequency division multiplexing (FDM) mode in which the sensor harvests RF energy and transmits information simultaneously over orthogonal frequency bands. The other is the time division multiplexing (TDM) mode in which energy harvesting and information transmission are implemented in the same frequency band but in different time slots. The energy harvesting channel and the information transmission channel are assumed to follow the Rician and the Rayleigh distributions, respectively, and are discretized and modeled as finite-state Markov chains. We formulate the process of energy harvesting and information transmission as an infinite-horizon discounted Markov decision process. The value iteration algorithm is used to find an asymptotically optimal energy harvesting and information transmission policy to optimize the long-term throughput. In the asymptotically optimal policy of the FDM mode, the energy transmitted from the sensor in one slot is proved to be non-decreasing with the battery state of the sensor. By contrast, such monotonicity between the transmitted energy and the battery state does not exist in the asymptotically optimal policy in the TDM mode. Simulation results verify the above findings and demonstrate that the proposed method outperforms the heuristic greedy method.