Prey Predator Fishery Model with Stage Structure in Two Patchy Marine Aquatic Environment

Author(s)
Mellachervu Naga Srinivas,
Mantripragada Ananata Satya Srinivas,
Kalyan Das,
Nurul Huda Gazi

ABSTRACT

In this paper, we propose and analyze a mathematical model to study the dynamics of a fishery resource system with stage structure in an aquatic environment that consists of two zones namely unreserved zone (fishing permitted) and reserved zone (fishing is strictly prohibited). In this model we introduce a stage structure in which predators are split into two kinds as immature predators and mature predators. It is assumed that immature predators cannot catch the prey and their foods are given by their parents (mature predators). It is also assumed that the fishing of immature predators prohibited in the unreserved zone and predator species are not allowed to enter inside the reserved zone. The local and global stability analysis has been specified. Biological and Bionomical equilibriums of the system are derived. Mathematical formulation of the optimal harvesting policy is given and its solution is derived in the equilibrium case by using Pontryagin’s maximum principle.

In this paper, we propose and analyze a mathematical model to study the dynamics of a fishery resource system with stage structure in an aquatic environment that consists of two zones namely unreserved zone (fishing permitted) and reserved zone (fishing is strictly prohibited). In this model we introduce a stage structure in which predators are split into two kinds as immature predators and mature predators. It is assumed that immature predators cannot catch the prey and their foods are given by their parents (mature predators). It is also assumed that the fishing of immature predators prohibited in the unreserved zone and predator species are not allowed to enter inside the reserved zone. The local and global stability analysis has been specified. Biological and Bionomical equilibriums of the system are derived. Mathematical formulation of the optimal harvesting policy is given and its solution is derived in the equilibrium case by using Pontryagin’s maximum principle.

KEYWORDS

Prey Predator, Stage Structure, Local and Global Stability, Bionomic Equilibrium, Optimal Harvesting, Pontryagin’s Maximum Principle

Prey Predator, Stage Structure, Local and Global Stability, Bionomic Equilibrium, Optimal Harvesting, Pontryagin’s Maximum Principle

Cite this paper

nullM. Srinivas, M. Srinivas, K. Das and N. Gazi, "Prey Predator Fishery Model with Stage Structure in Two Patchy Marine Aquatic Environment,"*Applied Mathematics*, Vol. 2 No. 11, 2011, pp. 1405-1416. doi: 10.4236/am.2011.211199.

nullM. Srinivas, M. Srinivas, K. Das and N. Gazi, "Prey Predator Fishery Model with Stage Structure in Two Patchy Marine Aquatic Environment,"

References

[1] F. Brauer and A. C. Soudack, “Stability Regions and Transition Phenomena for Harvested Predator-Prey Systems,” Journal of Mathematical Biology, Vol. 7, No. 4, 1979, pp. 319-337. doi:10.1007/BF00275152

[2] F. Brauer and A. C. Soudack, “Stability Regions in Predator-Prey Systems with Constant-Rate Prey Harvesting,” Journal of Mathematical Biology, Vol. 8, No. 1, 1979, pp. 55-71. doi:10.1007/BF00280586

[3] G. Dai and M. Tang, “Coexistence Region and Global Dynamics of a Harvested Predator-Prey System,” SIAM: SIAM Journal on Applied Mathematics, Vol. 58, No. 1, 1998, pp.193-210. doi:10.1137/S0036139994275799

[4] M. R. Myerscough, B. E. Gray, W. L. Hograth and J. Norbury, “An Analysis of an Ordinary Differential Equation Model for a Two-Species Predator-Prey System with Harvesting and Stocking,” Journal of Mathematical Biology, Vol. 30, 1992, pp. 389-401.

[5] K. S. Chaudhuri and S. S. Ray, “On the Combined Harvesting of a Prey-Predator System,” Journal of Biological Systems, Vol. 4, No. 3, 1996, pp. 373-389. doi:10.1142/S0218339096000259

[6] A. W. Leung, “Optimal Harvesting Co-Efficient Control of Steady State Prey-Predator Diffusive Volterra-Lotka Systems,” Applied Mathematics & Optimization, Vol. 31, No. 2, 1995, pp 219-241. doi:10.1007/BF01182789

[7] C. W. Clark, “Mathemalical Bioeconomics: The Optimal Management of Renewable Resources,” John Wiley and Sons, New York, 1979.

[8] S. A. Levin, T. G. Hallam and J. L. Gross, “Applied Mathematical Ecology,” Springer-Verlag, Berlin, 1989.

[9] W. G. Aiello and H. I. Freedman, “A Time Delay Model of Single Species Growth with Stage Structure,” Mathematical Biosciences, Vol. 101, No. 2, 1990, pp. 139-153. doi:10.1016/0025-5564(90)90019-U

[10] H. I. Freedman and K. Gopalsammy, “Global Stability in Time-Delayed Single Species Dynamics,” Bulletin of Mathematical Biology, Vol. 48, No. 5-6, 1986, pp. 485-492.

[11] G. Rosen, “Time Delays Produced by Essential Nonlinearity in Population Growth Models,” Bulletin of Mathematical Biology, Vol. 49, No. 2, 1987, pp. 253-255.

[12] M. E. Fisher and B. S. Goh, “Stability Results for Delayed Recruitment Models in Population Dynamics,” Journal of Mathematical Biology, Vol. 19, No. 1, 1984, pp. 147-156. doi:10.1007/BF00275937

[13] M. Mesterton-Gibbons, “On the Optimal Policy for the Combined Harvesting of Predator and Prey,” Natural Resource Modeling, Vol. 3, 1988, pp. 63-90.

[14] C. W. Clark, “Mathematical Bioeconomics: The Optimal Management of Renewable Resources,” Wiley, New York, 1976.

[15] K. S. Chaudhuri, “A Bio Economic Model of Harvesting of a Multi Species Fishery,” Ecological Modelling, Vol. 32, No. 4, 1986, pp. 267-279. doi:10.1016/0304-3800(86)90091-8

[16] D. L. Ragozin and G. Brown, “Harvest Policies and Non Market Valuation in a Predator Prey System,” Journal of Environmental Economics and Management, Vol. 12, No. 2, 1985, pp. 155-168. doi:10.1016/0095-0696(85)90025-7

[17] A. Hastings, “Global Stability of Two Species Systems,” Journal of Mathematical Biology, Vol. 5, 1978, pp.399-403.

[18] X.-Z. He, “Stability and Delays in a Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 198, No. 2, 1996, pp. 355-370. doi:10.1006/jmaa.1996.0087

[19] B. S. Goh, “Global Stability in Two Species Interactions,” Journal of Mathematical Biology, Vol. 3, No. 3-4, 1976, pp. 313-318. doi:10.1007/BF00275063

[20] W. G. Aiello and H. I. Freedman, “A Time Delay Model of Single Species Growth with Stage Structure,” Mathematical Biosciences, Vol. 101, No. 2, pp. 139-153. doi:10.1016/0025-5564(90)90019-U

[21] W. G. Aiello, H. I. Freedman and J. Wu, “Analysis of a Model Representing Stage Structured Population Growth with State-Dependent Time Delay,” SIAM: SIAM Journal on Applied Mathematics, Vol. 52, No. 3, 1992, pp. 855-869.

[22] T. K. Kar and M. Swarnakamal, “Influence of Prey Reserve in a Prey-Predator Fishery,” Non-Linear Analysis, Vol. 65, No. 9, 2006, pp.1725-1735.

[23] W. Wang and L. Chen, “Optimal Harvesting Policy for Single Population with Periodic Coefficients,” Mathematical Biosciences, Vol. 152, No. 2, 1998, pp. 165-177. doi:10.1016/S0025-5564(98)10024-X

[24] R. Zhang, J. F. Sun and H. X. Yang, “Analysis of a Prey-Predator Fishery Model with Prey Reserve,” Applied Mathematical Sciences, Vol. 50, No. 1, 2007, pp. 2481-2492.

[25] W. D. Wang, Y. Takeeuchi,Y. Saito and S. Nakaoka, “Prey-Predator System with Parental Care for Predators,” Journal of Theoritical Biology, Vol. 241, No. 3, 2005, pp. 451-458. doi:10.1016/j.jtbi.2005.12.008

[26] K. Das, N. H. Gazi, “Structural Stability Analysis of an Algal Bloom Mathematical Model in Trophic Interaction,” International Journal of Non-linear Ananlysis: Real World Applications, Vol. 11, No. 4, 2010, pp. 2191-2206.

[27] N. H. Gazi and K. Das, “Control of Parameters of a Delayed-Diffusive Autotroph-Herbivore System,” International Journal of Biological System, Vol. 18, No. 2, 2010, pp. 509-529.

[1] F. Brauer and A. C. Soudack, “Stability Regions and Transition Phenomena for Harvested Predator-Prey Systems,” Journal of Mathematical Biology, Vol. 7, No. 4, 1979, pp. 319-337. doi:10.1007/BF00275152

[2] F. Brauer and A. C. Soudack, “Stability Regions in Predator-Prey Systems with Constant-Rate Prey Harvesting,” Journal of Mathematical Biology, Vol. 8, No. 1, 1979, pp. 55-71. doi:10.1007/BF00280586

[3] G. Dai and M. Tang, “Coexistence Region and Global Dynamics of a Harvested Predator-Prey System,” SIAM: SIAM Journal on Applied Mathematics, Vol. 58, No. 1, 1998, pp.193-210. doi:10.1137/S0036139994275799

[4] M. R. Myerscough, B. E. Gray, W. L. Hograth and J. Norbury, “An Analysis of an Ordinary Differential Equation Model for a Two-Species Predator-Prey System with Harvesting and Stocking,” Journal of Mathematical Biology, Vol. 30, 1992, pp. 389-401.

[5] K. S. Chaudhuri and S. S. Ray, “On the Combined Harvesting of a Prey-Predator System,” Journal of Biological Systems, Vol. 4, No. 3, 1996, pp. 373-389. doi:10.1142/S0218339096000259

[6] A. W. Leung, “Optimal Harvesting Co-Efficient Control of Steady State Prey-Predator Diffusive Volterra-Lotka Systems,” Applied Mathematics & Optimization, Vol. 31, No. 2, 1995, pp 219-241. doi:10.1007/BF01182789

[7] C. W. Clark, “Mathemalical Bioeconomics: The Optimal Management of Renewable Resources,” John Wiley and Sons, New York, 1979.

[8] S. A. Levin, T. G. Hallam and J. L. Gross, “Applied Mathematical Ecology,” Springer-Verlag, Berlin, 1989.

[9] W. G. Aiello and H. I. Freedman, “A Time Delay Model of Single Species Growth with Stage Structure,” Mathematical Biosciences, Vol. 101, No. 2, 1990, pp. 139-153. doi:10.1016/0025-5564(90)90019-U

[10] H. I. Freedman and K. Gopalsammy, “Global Stability in Time-Delayed Single Species Dynamics,” Bulletin of Mathematical Biology, Vol. 48, No. 5-6, 1986, pp. 485-492.

[11] G. Rosen, “Time Delays Produced by Essential Nonlinearity in Population Growth Models,” Bulletin of Mathematical Biology, Vol. 49, No. 2, 1987, pp. 253-255.

[12] M. E. Fisher and B. S. Goh, “Stability Results for Delayed Recruitment Models in Population Dynamics,” Journal of Mathematical Biology, Vol. 19, No. 1, 1984, pp. 147-156. doi:10.1007/BF00275937

[13] M. Mesterton-Gibbons, “On the Optimal Policy for the Combined Harvesting of Predator and Prey,” Natural Resource Modeling, Vol. 3, 1988, pp. 63-90.

[14] C. W. Clark, “Mathematical Bioeconomics: The Optimal Management of Renewable Resources,” Wiley, New York, 1976.

[15] K. S. Chaudhuri, “A Bio Economic Model of Harvesting of a Multi Species Fishery,” Ecological Modelling, Vol. 32, No. 4, 1986, pp. 267-279. doi:10.1016/0304-3800(86)90091-8

[16] D. L. Ragozin and G. Brown, “Harvest Policies and Non Market Valuation in a Predator Prey System,” Journal of Environmental Economics and Management, Vol. 12, No. 2, 1985, pp. 155-168. doi:10.1016/0095-0696(85)90025-7

[17] A. Hastings, “Global Stability of Two Species Systems,” Journal of Mathematical Biology, Vol. 5, 1978, pp.399-403.

[18] X.-Z. He, “Stability and Delays in a Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 198, No. 2, 1996, pp. 355-370. doi:10.1006/jmaa.1996.0087

[19] B. S. Goh, “Global Stability in Two Species Interactions,” Journal of Mathematical Biology, Vol. 3, No. 3-4, 1976, pp. 313-318. doi:10.1007/BF00275063

[20] W. G. Aiello and H. I. Freedman, “A Time Delay Model of Single Species Growth with Stage Structure,” Mathematical Biosciences, Vol. 101, No. 2, pp. 139-153. doi:10.1016/0025-5564(90)90019-U

[21] W. G. Aiello, H. I. Freedman and J. Wu, “Analysis of a Model Representing Stage Structured Population Growth with State-Dependent Time Delay,” SIAM: SIAM Journal on Applied Mathematics, Vol. 52, No. 3, 1992, pp. 855-869.

[22] T. K. Kar and M. Swarnakamal, “Influence of Prey Reserve in a Prey-Predator Fishery,” Non-Linear Analysis, Vol. 65, No. 9, 2006, pp.1725-1735.

[23] W. Wang and L. Chen, “Optimal Harvesting Policy for Single Population with Periodic Coefficients,” Mathematical Biosciences, Vol. 152, No. 2, 1998, pp. 165-177. doi:10.1016/S0025-5564(98)10024-X

[24] R. Zhang, J. F. Sun and H. X. Yang, “Analysis of a Prey-Predator Fishery Model with Prey Reserve,” Applied Mathematical Sciences, Vol. 50, No. 1, 2007, pp. 2481-2492.

[25] W. D. Wang, Y. Takeeuchi,Y. Saito and S. Nakaoka, “Prey-Predator System with Parental Care for Predators,” Journal of Theoritical Biology, Vol. 241, No. 3, 2005, pp. 451-458. doi:10.1016/j.jtbi.2005.12.008

[26] K. Das, N. H. Gazi, “Structural Stability Analysis of an Algal Bloom Mathematical Model in Trophic Interaction,” International Journal of Non-linear Ananlysis: Real World Applications, Vol. 11, No. 4, 2010, pp. 2191-2206.

[27] N. H. Gazi and K. Das, “Control of Parameters of a Delayed-Diffusive Autotroph-Herbivore System,” International Journal of Biological System, Vol. 18, No. 2, 2010, pp. 509-529.