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 Analisis Kestabilan dan Keuntungan Maksimum Model Predator-Prey Fungsi Respon Tipe Holling III dengan Usaha Pemanenan

2018 ◽  
Author(s):  
DIDIHARYONO DIDIHARYONO
Keyword(s):  
Equilibrium Point ◽  
Linearization Method ◽  
Stability Test ◽  
Optimal Harvesting ◽  
Predator Prey ◽  
Type Iii ◽  
Hurwitz Stability ◽  
Predator Prey Model ◽  
Maximum Profit ◽  
Prey Model

In this paper, we discussed stability analysis of predator-prey model with Holling type III and will harvesting effort at second populations. The research aimed is, to investigate solution the predator-prey model with Holling type III functional response with addition harvesting effort and to investigate maximum profit from optimal harvesting at second populations. Stability of equilibrium point use linearization method and determine the stability by notice the eigenvalues of Jacoby matrix evaluation of equilibrium point and can also be determined using Hurwitz stability test by observing the coefficients of the characteristic equation. The result shows that the obtained an interior point 〖TE〗_2^* (x^*,y^*) which asymptotic stable according to Hurwitz stability test and find maximum profit of exploitation effort or harvest at second populations. Predator-prey population is always exist in their life, although exploitation with harvesting effort and given maximum profit is π_max=162.68 where to find maximum profit on critical points of surface profit function.

BIFURCATION ANALYSIS OF A TWO-DIMENSIONAL PREDATOR–PREY MODEL WITH HOLLING TYPE IV FUNCTIONAL RESPONSE AND NONLINEAR PREDATOR HARVESTING

2020 ◽  
Vol 28 (04) ◽  
pp. 839-864
Author(s):  
UTTAM GHOSH ◽  
PRAHLAD MAJUMDAR ◽  
JAYANTA KUMAR GHOSH
Keyword(s):  
Equilibrium Point ◽  
Functional Response ◽  
Conversion Efficiency ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
Trivial Equilibrium

The aim of this paper is to investigate the dynamical behavior of a two-species predator–prey model with Holling type IV functional response and nonlinear predator harvesting. The positivity and boundedness of the solutions of the model have been established. The considered system contains three kinds of equilibrium points. Those are the trivial equilibrium point, axial equilibrium point and the interior equilibrium points. The trivial equilibrium point is always saddle and stability of the axial equilibrium point depends on critical value of the conversion efficiency. The interior equilibrium point changes its stability through various parametric conditions. The considered system experiences different types of bifurcations such as Saddle-node bifurcation, Hopf bifurcation, Transcritical bifurcation and Bogdanov–Taken bifurcation. It is clear from the numerical analysis that the predator harvesting rate and the conversion efficiency play an important role in stability of the system.

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

Flip bifurcation and Neimark–Sacker bifurcation in a discrete predator–prey model with harvesting

2019 ◽  
Vol 13 (01) ◽  
pp. 1950093
Author(s):  
Wei Liu ◽  
Yaolin Jiang
Keyword(s):  
Equilibrium Point ◽  
Discrete Dynamical System ◽  
Singular System ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

In this paper, a difference-algebraic predator–prey model is proposed, and its complex dynamical behaviors are analyzed. The model is a discrete singular system, which is obtained by using Euler scheme to discretize a differential-algebraic predator–prey model with harvesting that we establish. Firstly, the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory. Further, by applying the new normal form of difference-algebraic equations, center manifold theory and bifurcation theory, the Flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point are studied, where the step size is treated as the variable bifurcation parameter. Lastly, with the help of Matlab software, some numerical simulations are performed not only to validate our theoretical results, but also to show the abundant dynamical behaviors, such as period-doubling bifurcations, period 2, 4, 8, and 16 orbits, invariant closed curve, and chaotic sets. In particular, the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.

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