Optimal harvesting of a two species competition model with imprecise biological parameters

In this paper a two competing species harvesting model with imprecise biological parameters has been developed. We have developed a method to handle these imprecise parameters and discuss the dynamical behaviour of the model. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. Also the bionomic equilibrium of the harvesting model has been analysed. Next the equilibrium solution of the control problem has been derived, and then dynamical optimization of the harvest policy is carried out taking combined harvesting effort as a dynamic variable by invoking Pontryagin’s Maximum Principle. Our important analytical findings are illustrated through computer simulation using MATLAB followed by discussions and conclusions.

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

[2]  Shigui Ruan,et al.  Global Analysis in a Predator-Prey System with Nonmonotonic Functional Response , 2001, SIAM J. Appl. Math..

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

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

[5]  H. I. Freedman,et al.  A mathematical model of facultative mutualism with populations interacting in a food chain. , 1989, Mathematical biosciences.

[6]  P D N Srinivasu,et al.  Biological control through provision of additional food to predators: a theoretical study. , 2007, Theoretical population biology.

[7]  V. Sree Hari Rao,et al.  Three-species food-chain models with mutual interference and time delays , 1986 .

[8]  G. Leitmann.  An Introduction To Optimal Control , 1966 .

[9]  H. I. Freedman,et al.  Mathematical analysis of some three-species food-chain models , 1977 .

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

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

[12]  G. Samanta,et al.  Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model. , 2013, Mathematical biosciences.

[13]  P. Verhulst,et al.  Notice sur la loi que la population suit dans son accroissement. Correspondance Mathematique et Physique Publiee par A , 1838 .

[14]  H. I. Freedman,et al.  Persistence in models of three interacting predator-prey populations , 1984 .

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

[16]  Rong Yuan,et al.  Dynamics in two nonsmooth predator–prey models with threshold harvesting , 2013 .

[17]  Yang Kuang,et al.  Simple Food Chain in a Chemostat with Distinct Removal Rates , 2000 .

[18]  M. Mesterton-Gibbons.  ON THE OPTIMAL POLICY FOR COMBINED HARVESTING OF INDEPENDENT SPECIES , 1987 .

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

[20]  Y. Takeuchi,et al.  Persistence and periodic orbits of a three-competitor model with refuges. , 1992, Mathematical biosciences.

[21]  Yi Shen,et al.  Hopf bifurcation of a predator–prey system with predator harvesting and two delays , 2013 .

[22]  S Rinaldi,et al.  Remarks on food chain dynamics. , 1996, Mathematical biosciences.

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

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

[25]  B. Goh,et al.  Management and analysis of biological populations , 1982 .

[26]  Yong Xue,et al.  Spatiotemporal dynamics of a predator–prey model , 2011, Nonlinear Dynamics.

[27]  R. Yedavalli,et al.  Robust Stability and Control of Linear Interval Parameter Systems Using Quantitative (State Space) and Qualitative (Ecological) Perspectives , 2011 .

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

[29]  M E Gilpin,et al.  Enriched predator-prey systems: theoretical stability. , 1972, Science.

[30]  Mike Mesterton-Gibbons.  ON THE OPTIMAL POLICY FOR COMBINING HARVESTING OF PREDATOR AND PREY , 1988 .

[31]  J. Wilen,et al.  Optimal recovery paths for perturbations of trophic level bioeconomic systems , 1986 .

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

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

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

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

[36]  Mark Kot,et al.  Elements of Mathematical Ecology: Frontmatter , 2001 .

[37]  H. I. Freedman,et al.  Persistence in a model of three competitive populations , 1985 .

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

[39]  R. Levins.  The strategy of model building in population biology , 1966 .

[40]  T. Malthus.  An essay on the principle of population, as it affects the future improvement of society , 2006 .

[41]  Alakes Maiti,et al.  Effect of time-delay on a food chain model , 2008, Appl. Math. Comput..

[42]  Alakes Maiti,et al.  Deterministic and stochastic analysis of a prey-dependent predator–prey system , 2005 .

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

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

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

[46]  Li Li,et al.  Pattern dynamics of a spatial predator–prey model with noise , 2012 .