Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model.

The paper presents the study of one prey one predator harvesting model with imprecise biological parameters. 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. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin's maximum principle. Numerical examples are presented to support the proposed model.

[1]  Colin W. Clark,et al.  Mathematical Bioeconomics: The Optimal Management of Renewable Resources. , 1993 .

[2]  Eric Nævdal,et al.  Optimal management of renewable resources with Darwinian selection induced by harvesting , 2008 .

[3]  Chung-Chiang Chen,et al.  Fishery policy when considering the future opportunity of harvesting. , 2007, Mathematical biosciences.

[4]  E. Blum,et al.  The Mathematical Theory of Optimal Processes. , 1963 .

[5]  Peeyush Chandra,et al.  A model for fishery resource with reserve area , 2003 .

[6]  R. Bassanezi,et al.  Fuzzy modelling in population dynamics , 2000 .

[7]  John Pastor,et al.  Mathematical Ecology of Populations and Ecosystems , 2008 .

[8]  Xiaoping Xue,et al.  Impulsive functional differential inclusions and fuzzy population models , 2003, Fuzzy Sets Syst..

[9]  D. Ragozin,et al.  Harvest policies and nonmarket valuation in a predator -- prey system , 1985 .

[10]  Laécio C. Barros,et al.  Attractors and asymptotic stability for fuzzy dynamical systems , 2000, Fuzzy Sets Syst..

[11]  Christophe Béné,et al.  A viability analysis for a bio-economic model , 2001 .

[12]  K. S. Chaudhuri,et al.  ON THE COMBINED HARVESTING OF A PREY-PREDATOR SYSTEM , 1996 .

[13]  A. J. Lotka,et al.  Elements of Physical Biology. , 1925, Nature.

[14]  G. S. Mahapatra,et al.  Posynomial Parametric Geometric Programming with Interval Valued Coefficient , 2012, Journal of Optimization Theory and Applications.

[15]  Rodney Carlos Bassanezi,et al.  Predator–prey fuzzy model , 2008 .

[16]  Colin W. Clark,et al.  Bioeconomic Modelling and Fisheries Management. , 1985 .

[17]  D K Bhattacharya,et al.  Bionomic equilibrium of two-species system. I. , 1996, Mathematical biosciences.

[18]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[19]  K. S. Chaudhuri,et al.  Harvesting of a prey–predator fishery in the presence of toxicity , 2009 .

[20]  Alakes Maiti,et al.  Bioeconomic modelling of a three-species fishery with switching effect , 2003 .

[21]  R. Hannesson.  Optimal harvesting of ecologically interdependent fish species , 1983 .

[22]  K. S. Chaudhuri,et al.  Dynamic optimization of combined harvesting of a two-species fishery , 1988 .

[23]  Ke Wang,et al.  Optimal harvesting policy for general stochastic Logistic population model , 2010 .

[24]  A. J. Lotka.  Elements of Physical Biology. , 1925, Nature.

[25]  Eduardo González-Olivares,et al.  Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response , 2012 .

[26]  G. S. Mahapatra,et al.  Reliability and cost analysis of series system models using fuzzy parametric geometric programming , 2010 .

[27]  Shujing Gao,et al.  Harvesting of a phytoplankton–zooplankton model , 2010 .

[28]  Huan Su,et al.  Optimal harvesting policy for stochastic Logistic population model , 2011, Appl. Math. Comput..

[29]  Jorge Rebaza,et al.  Dynamics of prey threshold harvesting and refuge , 2012, J. Comput. Appl. Math..

[30]  Liu Hsu,et al.  Achieving global convergence to an equilibrium population in predator-prey systems by the use of a discontinuous harvesting policy , 2000 .