scholarly journals Optimal control for a size-structured predator-prey model in a polluted environment

2021 ◽  
Vol 269 ◽  
pp. 01004
Author(s):  
Tainian Zhang ◽  
Zhixue Luo
Keyword(s):  
Size Structure ◽  
Necessary Optimality Conditions ◽  
Optimal Harvesting ◽  
Polluted Environment ◽  
Nonlinear Functional ◽  
Predator Prey ◽  
Optimal Harvesting Policy ◽  
Predator Prey Model ◽  
Dependent Model ◽  
The Fixed Point Theorem

In this paper, we deal with an optimal harvesting problem for a periodic predator-prey hybrid system dependent on size-structure in a polluted environment. In other words, a size-dependent model in an environment with a small toxicant content has been established. The well-posedness of state system is proved by using the fixed point theorem. The necessary optimality conditions are derived by tangent-normal cone technique in nonlinear functional analysis. The existence of a unique optimal harvesting policy is verified via the Ekeland’s variational principle. The optimal harvesting problem has an optimal harvesting policy, which has a Bang-Bang structure and provides a threshold for the optimal harvesting problem. Using the optimization theories and methods in mathematics to control phenomena of life. The objective function represents the total economic profit from the harvested population. Some theoretical results obtained in this paper provide a scientific theoretical basis for the practical application of the model.

scholarly journals Dynamic Behaviors of a Harvesting Leslie-Gower Predator-Prey Model

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Na Zhang ◽  
Fengde Chen ◽  
Qianqian Su ◽  
Ting Wu
Keyword(s):  
Sufficient Conditions ◽  
Practical Significance ◽  
Optimal Harvesting ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Optimal Harvesting Policy ◽  
Predator Prey Model ◽  
Prey Model ◽  
Suitable Restriction

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 thePontryagin'smaximal principle. At last, an example is given to show that the optimal harvesting policy is realizable.

The stage-structured predator–prey model and optimal harvesting policy

2000 ◽  
Vol 168 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Xin-an Zhang ◽  
Lansun Chen ◽  
Avidan U Neumann
Keyword(s):  
Optimal Harvesting ◽  
Predator Prey ◽  
Optimal Harvesting Policy ◽  
Predator Prey Model ◽  
Prey Model

scholarly journals Optimal Harvesting Policy of Predator-Prey Model with Free Fishing and Reserve Zones

2017 ◽  
Author(s):  
Syamsuddin Toaha ◽  
Rustam Rustam
Keyword(s):  
Equilibrium Point ◽  
Optimal Harvesting ◽  
Present Value ◽  
Predator Prey ◽  
Optimal Harvesting Policy ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Maximum Profit ◽  
Prey Model ◽  
Net Revenue

If any question related to this paper, please ask via [email protected]. This article is an INA-Rxiv post-print post from the OSF (Open Science Framework) which is self-archiving. This article has been presented on SYMPOSIUM ON BIOMATHEMATICS (SYMOMATH 2016) and its proceedings are published in AIP Conference Proceedings, Volume 1825, Issue 1 with links http://aip.scitation.org/doi/abs/10.1063/1.4978992The present paper deals with an optimal harvesting of predator-prey model in an ecosystem that consists of two zones, namely the free fishing and prohibited zones. The dynamics of prey population in the ecosystem can migrate from the free fishing to the prohibited zone and vice versa. The predator and prey populations in the free fishing zone are then harvested with constant efforts. The existence of the interior equilibrium point is analyzed and its stability is determined using Routh-Hurwitz stability test. The stable interior equilibrium point is then related to the problem of maximum profit and the problem of present value of net revenue. We follow the Pontryagin’s maximal principle to get the optimal harvesting policy of the present value of the net revenue. From the analysis, we found a critical point of the efforts that makes maximum profit. There also exists certain conditions of the efforts that makes the present value of net revenue becomes maximal. In addition, the interior equilibrium point is locally asymptotically stable which means that the optimal harvesting is reached and the unharvested prey, harvested prey, and harvested predator populations remain sustainable. Numerical examples are given to verify the analytical results.

scholarly journals A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aziz Khan ◽  
Hashim M. Alshehri ◽  
J. F. Gómez-Aguilar ◽  
Zareen A. Khan ◽  
G. Fernández-Anaya
Keyword(s):  
Fractional Order ◽  
Existence And Uniqueness ◽  
Variable Order ◽  
Fractional Order Systems ◽  
New Approach ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fractional Differential ◽  
The Fixed Point Theorem

AbstractThis paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method.

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

2020 ◽  
Vol 18 (1) ◽  
pp. 458-475
Author(s):  
Na Zhang ◽  
Yonggui Kao ◽  
Fengde Chen ◽  
Binfeng Xie ◽  
Shiyu Li
Keyword(s):  
Water Level ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Optimal Harvesting ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Model Interaction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fluctuating Water Level

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.

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