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Effects of Over-Harvesting and Drought on a Predator-Prey System with Optimal Control

DOI: 10.4236/oje.2018.88028, PP. 459-482

Keywords: Predator-Prey System, Over-Harvesting, Drought, Optimal Control

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In this paper, a two species predator-prey model is developed where prey is affected by over-harvesting and drought and predator is affected by drought. The intention is to investigate the impact of over-harvesting and drought on predator-prey system, and suggest control strategies to alleviate the problem of loss of prey and predator species due to over-harvesting and drought. The control strategies suggested are creation of reserve areas with restriction of harvesting for controlling over-harvesting and construction of dams for mitigating drought effects. The results obtained from theoretical and numerical simulation of the predator-prey model with harvesting and drought without control strategies showed that, both harvesting and drought affect the predator-prey population negatively. However, the results obtained from numerical simulations of the model with control measures showed that, the use of control strategies one at a time encourages the increase of the prey and predator species to the optimal population size. Furthermore, the best result is obtained when control strategies, creation of reserve areas with restriction of harvesting and construction of dams are applied simultaneously.


[1]  Begon, M., Townsend, C.R. and Harper, J.L. (2006) ECOLOGY: From Individuals to Ecosystem. CPI Bath Press, UK.
[2]  Taylor, R.J. (1984) Predation. Champman and Hall, New York.
[3]  Berryman, A.A. (1992) The Origins and Evolution of Predator Prey Theory. Ecology, 73, 1530-1535.
[4]  Dubey, B. (2007) A Prey-Predator Model with a Reserved Area. Nonlinear Analysis: Modeling and Control, 12, 479-494.
[5]  Sagamiko, T.D., Shaban, N., Nahonyo, C.L. and Makinde, O.D. (2015) Optimal Control of a Threatened Wildebeest-Lion Prey-Predator System in the Serengeti Ecosystem. Open Journal of Ecology, 5, 110-119.
[6]  Sperry, J.H. and Weatherhead, P.J. (2008) Prey-Mediated Effects of Drought on Condition and Survival of a Terrestrial Snake. Ecological Society of America, 89, 2770-2776.
[7]  Chakraborty, K., Das, S. and Kar, T.K. (2011) Optimal Control of Effort of a Stage Structured Prey Predator Fishery Model with Harvesting. Nonlinear Analysis: Real World Applications, 12, 3452-3467.
[8]  Chakraborty, K., Chakraborty, M. and Kar, T.K. (2011) Optimal Control of Harvesting and Bifurcation of a Prey-Predator Model with Stage Structure. Applied Mathematics and Computation, 217, 8778-8792.
[9]  Kaiyuan, L. and Lansun, C. (2007) Harvesting Control for a Stage-Structured Predator-Prey Model with Ivlev’s Functional Response and Impulsive Stocking on Prey. Discrete Dynamics in nature and Society, 2007, Article ID: 86482.
[10]  Kar, T.K. and Ghosh, B. (2010) Bifurcations and Feedback Control of a Stage Structure Exploited Prey Predator System. International Journal of Engineering, Science and Technology, 2, 131-141.
[11]  Hanson, P.J. and Weltzin, J.F. (2000) Drought Disturbance from Climate Change: Response of United States Forests. Science of the Total Environment, 262, 205-220.
[12]  Pickett, S.T.A. and White, P.S. (1985) The Ecology of Natural Disturbance and Patch Dynamics. Academic Press, London.
[13]  Pickett, S.T.A., Wu, J.G. and Cadenasso, M.L. (1999) Patch Dynamics and the Ecology of Disturbed Ground. Elsevier, Amsterdam, 707-722.
[14]  Fernandez-Cara, E. and ZuaZua, E. (2003) Control Theory: History, Mathematical Achievements and Perspectives. SeMA Journal, 26, 79-140.
[15]  Bera, S.P., Maiti, A. and Samanta, G.P. (2014) A Prey-Predator Model with Infection in Both Prey and Predator. Filomat, 29,1753-1767.
[16]  Kar, T. and Ghosh, B. (2012) Sustainability and Optimal Control of an Exploited Prey-Predator System through Provision of Alternative Food to Predator. Biosystems, 109, 220-232.
[17]  Hugo, A., Massawe, E.S. and Makinde, O.D. (2012) An Eco-Epidemiological Mathematical Model with Treatment and Disease Infection in both Prey and Predator Population. Journal of Ecology and the Natural Environment, 4, 266-279.
[18]  Bodine, E., Gross, L. and Lenhart, S. (2008) Optimal Control Applied to a Model for Species Augmentation. Mathematical Bioscience and Engineering, 5, 669-680.
[19]  Goh, B., Leitman, G. and Vicent, T.L. (1974) Optimal Control of a Prey Predator System. Mathematical Bioscience, 19, 263-286.
[20]  Naji, R.K. and Dina, S.A.J. (2011) The Dynamics of Stage Structured Prey-Predator Model Involving Parasitic Infectious Disease. Applied Mathematics: An International Journal, 6, 529-551.
[21]  Stoddart, A.W.J. (1967) Existence of Optimal Controls. Pacific Journal of Mathematics, 20, 167-177.
[22]  Fleming, W.H. and Rishel, R.W. (1975) Deterministic and Stochastic Optimal Control. Vol. 268, Springer-Verlag, New York.
[23]  Lenhart, S. and Workman, J. (2007) Optimal Control Applied to Biological Models. Champman and Hall/CRC, London.
[24]  Schaller, G.B. (1972) The Serengeti Lion: A Study of Predator-Prey Relations. University of Chicagopress, Chicago.
[25]  Kar, T.K. (2010) A Dynamic Reaction Model of a Prey-Predator System with Stage-Structure for Predator. Modern Applied Science, 4, 183-195.


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