Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem

Thadei Damas Sagamiko; Nyimvua Shaban; Cuthbert Leonard Nahonyo; Oluwole Daniel Makinde

doi:doi:10.11648/j.acm.20150404.18

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Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem

Thadei Damas Sagamiko, Nyimvua Shaban, Cuthbert Leonard Nahonyo, Oluwole Daniel Makinde

Published in Applied and Computational Mathematics (Volume 4, Issue 4)

Received: 29 June 2015 Accepted: 9 July 2015 Published: 17 July 2015

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Abstract

In this paper a two species prey-predator model is developed in which prey is wildebeest and predator is lion and both are threatened by poaching, drought and diseases.The system is found in the Serengeti ecosystem.The model is constructed based on Holling type II functional response incorporating a constant prey refuge. We apply optimal control theory to investigate optimal strategies for controlling the threats in the system where anti-poaching patrols are used for controlling poaching, construction of dams for mitigating drought and vaccination for diseases control. The possible impact of using combinations of three controls either one at a time or two at a time on the threatened system plus a refuge factor is examined. All control strategies have shown significant increase in prey and predator populations . However, the best result is achieved by controlling all threats together. The effect of variation of prey refuge to the control of threats is studied and results indicate that increase of causes more prey individuals to be saved and reduces the number of predator individuals saved. This behaviour agrees with theoretical results obtained in co-existence equilibrium point.

Published in	Applied and Computational Mathematics (Volume 4, Issue 4)
DOI	10.11648/j.acm.20150404.18
Page(s)	296-312
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Optimal Control, Prey-predator System, Prey Refuge, Threat, Gregariousness

References

[1]	Agnihotri, K.L. ( 2012). The dynamics of disease transmission in a preypredator system with harvesting of prey. International Journal of Advanced Research in Computer Engineering and Technology, 1.
[2]	Bolger, D., Newmark ,W., Morrison, T., and Doak ,D. (2008). The need for integrative approaches to understand and conserve migratory ungulates. Ecology Letters , 11: 63-77.
[3]	Borner, M. (1995). The great Migration [Online article]: cited on 15th May 2013, http://www.serengeti.org/download/migration.pdf
[4]	Chakraborty, K., and Das, S.S.( 2014) . Biological conservation of a prey-predator system incorporating constant prey refuge through provision of alternative food to predators: A theoretical study, Acta Biotheor, 62:183-205.
[5]	Fleming, W.H., and Rishel, R.W.( 1975). Deterministic and stochastic optimal control, 268, Springer-Verlag New York.
[6]	Fryxell, J.M., Mosser, A., Sinclair, A.R.E., and Packer, C.( 2007). Group formation stabilizes predator-prey dynamics. Nature ,449.
[7]	GEAS. (2013) Saving the Great migrations: Declining Wildebeest in East Africa? [Online article]:cited on 28th September 2014, http://na.unep.net/geas/getUNEPPageWithArticleIDScript.php?article_id=107
[8]	Gonzalez-Olivares, E., and Ramos-Jiliberto, R.( 2003). Dynamic consenquences of refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecological modelling ,166 :135-146.
[9]	Haris, G., Thirgood, S., Hopcraft, J., Cromsigt, J.,and Berger, J.( 2009). Global decline in aggregated migrations of large terrestrial mammals. Endangered species Research, 7: 55-76.
[10]	Holling, C.S.( 1965). The functional Response of Predators to Prey density and its role in Mimicry and population regulation. Memoirs of the Entomological society of Canada ,97: 1-60.
[11]	Hopcraft. J., Sinclair, A., Holdo, R., Mwangomo, E., Mduma, S., Thirgood, S., Borner, M., Fryxell, J., and Olff, H.( 2013)Why are wildebeest the most abundant herbivore in the Serengeti ecosystem?, Serengeti IV: Sustaining biodiversity in a coupled Human-Natureal system (University of Chicago press. Chicago, 2013)
[12]	Ikanda, D., and Packer, C.( 2008). Ritual vs. Retaliatory killing of African lions in the Ngorongoro conservation Area, Tanzania. Endangered species Research, 6: 67-74.
[13]	Kar,T., and Gosh, B.( 2012). Sustainability and Optimal control of an exploited prey-predator system through provision of alternative food to predator. Biosystems, 109:220-232.
[14]	Kolowski, J.M., and Holekam, K.E.( 2006). Spatial, temporal and physical characteristics of Livestock depradations by large Carnivores along a Kenyan reserve border. Biol Conserv 128: 529-554.
[15]	Ma, Z., Li, W., Zhao, Y., Wang, W., Zhang, H., and Li, Z.( 2009). Effects of prey refuge on a predator-prey model with a class of functional response: The role of refuges. Matematical Biosciences 218: 73-79.
[16]	Mduma,S.R(1996). Serengeti Wildebeest population dynamics: Regulation, Limitation and Implications for harvesting, Ph.D. Thesis, University of British Columbia, Canada, 1996.
[17]	Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., and Mishchenko, E. The mathematical Theory of Optimal process. Wiley, New York, 1962.
[18]	Riipi, M., Alatalo, R.V., Lindstorm, L., and Mappes, J.( 2001). Multiple benefits of gregariousness cover detectability costs in aposematic aggregations. Nature, 413: 512-4.
[19]	Roelke-Parker, M., Munson, L., Palker, C., Kock, R., Cleaveland, S., Carpenter, M., Brien, S.J, Popischil, A., Hoffman-Lehmann, R., Lutz ,H., Mwamengele, G.L.N., Mgasa, M.N., Machange, G.A., Summers, B.A., and Appel, M.J.G.( 1996). A canine distemper virus epidemic in Serengeti lions (Panteraleo). Nature. 379: 441-445.
[20]	Schaller, G.B, The Serengeti Lion: A study of Predator-Prey relations. University of Chicagopress, Chicago, 1972.
[21]	Sih, A.( 1986). Antipredator responses and the perception of danger by mosquito larvae. Ecology, 67: 434-441.
[22]	Sih, A., Pentraka, J.W., and Kats, L.B.( 1988). The dynamics of prey refuge use: a model and tests with sunfish and salamanders larvae. The Am. Naturalist 132: 463-483.
[23]	Sillen-Tallberg, B., and Leimar, O. (1988). The evolution of gregariousness in Distasteful Insects as Defence against Predators. The American Naturalist, 132:723-734.
[24]	Sinclair, A.R.E., Packer, C., Mduma, S.A.R., and Fryxell, J.M. Serengeti III, Human impacts on Ecosytems dynamics, The University of Chicagopress, chicago, 2008.

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Thadei Damas Sagamiko, Nyimvua Shaban, Cuthbert Leonard Nahonyo, Oluwole Daniel Makinde. (2015). Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem. Applied and Computational Mathematics, 4(4), 296-312. https://doi.org/10.11648/j.acm.20150404.18

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ACS Style

Thadei Damas Sagamiko; Nyimvua Shaban; Cuthbert Leonard Nahonyo; Oluwole Daniel Makinde. Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem. Appl. Comput. Math. 2015, 4(4), 296-312. doi: 10.11648/j.acm.20150404.18

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AMA Style

Thadei Damas Sagamiko, Nyimvua Shaban, Cuthbert Leonard Nahonyo, Oluwole Daniel Makinde. Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem. Appl Comput Math. 2015;4(4):296-312. doi: 10.11648/j.acm.20150404.18

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@article{10.11648/j.acm.20150404.18,
  author = {Thadei Damas Sagamiko and Nyimvua Shaban and Cuthbert Leonard Nahonyo and Oluwole Daniel Makinde},
  title = {Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {4},
  pages = {296-312},
  doi = {10.11648/j.acm.20150404.18},
  url = {https://doi.org/10.11648/j.acm.20150404.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150404.18},
  abstract = {In this paper a two species prey-predator model is developed in which prey is wildebeest and predator is lion and both are threatened by poaching, drought and diseases.The system is found in the Serengeti ecosystem.The model is constructed based on Holling type II functional response incorporating a constant prey refuge. We apply optimal control theory to investigate optimal strategies for controlling the threats in the system where anti-poaching patrols are used for controlling poaching, construction of dams for mitigating drought and vaccination for diseases control. The possible impact of using combinations of three controls either one at a time or two at a time on the threatened system plus a refuge factor is examined. All control strategies have shown significant increase in prey and predator populations . However, the best result is achieved by controlling all threats together. The effect of variation of prey refuge   to the control of threats is studied and results indicate that increase of  causes more prey individuals to be saved and reduces the number of predator individuals saved. This behaviour agrees with theoretical results obtained in co-existence equilibrium point.},
 year = {2015}
}

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TY  - JOUR
T1  - Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem
AU  - Thadei Damas Sagamiko
AU  - Nyimvua Shaban
AU  - Cuthbert Leonard Nahonyo
AU  - Oluwole Daniel Makinde
Y1  - 2015/07/17
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150404.18
DO  - 10.11648/j.acm.20150404.18
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 296
EP  - 312
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150404.18
AB  - In this paper a two species prey-predator model is developed in which prey is wildebeest and predator is lion and both are threatened by poaching, drought and diseases.The system is found in the Serengeti ecosystem.The model is constructed based on Holling type II functional response incorporating a constant prey refuge. We apply optimal control theory to investigate optimal strategies for controlling the threats in the system where anti-poaching patrols are used for controlling poaching, construction of dams for mitigating drought and vaccination for diseases control. The possible impact of using combinations of three controls either one at a time or two at a time on the threatened system plus a refuge factor is examined. All control strategies have shown significant increase in prey and predator populations . However, the best result is achieved by controlling all threats together. The effect of variation of prey refuge   to the control of threats is studied and results indicate that increase of  causes more prey individuals to be saved and reduces the number of predator individuals saved. This behaviour agrees with theoretical results obtained in co-existence equilibrium point.
VL  - 4
IS  - 4
ER  -

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Author Information

Thadei Damas Sagamiko

Department of Mathematics, University of Dar Es Salaam, Dar Es Salaam, Tanzania
Nyimvua Shaban

Department of Mathematics, University of Dar Es Salaam, Dar Es Salaam, Tanzania
Cuthbert Leonard Nahonyo

Department of Zoology and Wildlife Conservation, University of Dar Es Salaam, Dar Es Salaam, Tanzania
Oluwole Daniel Makinde

Faculty of Military Science, Stellenbosch University, Stellenbosch, South Africa

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Thadei Damas Sagamiko, Nyimvua Shaban, Cuthbert Leonard Nahonyo, Oluwole Daniel Makinde. (2015). Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem. Applied and Computational Mathematics, 4(4), 296-312. https://doi.org/10.11648/j.acm.20150404.18

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ACS Style

Thadei Damas Sagamiko; Nyimvua Shaban; Cuthbert Leonard Nahonyo; Oluwole Daniel Makinde. Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem. Appl. Comput. Math. 2015, 4(4), 296-312. doi: 10.11648/j.acm.20150404.18

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AMA Style

Thadei Damas Sagamiko, Nyimvua Shaban, Cuthbert Leonard Nahonyo, Oluwole Daniel Makinde. Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem. Appl Comput Math. 2015;4(4):296-312. doi: 10.11648/j.acm.20150404.18

Copy | Download

@article{10.11648/j.acm.20150404.18,
  author = {Thadei Damas Sagamiko and Nyimvua Shaban and Cuthbert Leonard Nahonyo and Oluwole Daniel Makinde},
  title = {Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {4},
  pages = {296-312},
  doi = {10.11648/j.acm.20150404.18},
  url = {https://doi.org/10.11648/j.acm.20150404.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150404.18},
  abstract = {In this paper a two species prey-predator model is developed in which prey is wildebeest and predator is lion and both are threatened by poaching, drought and diseases.The system is found in the Serengeti ecosystem.The model is constructed based on Holling type II functional response incorporating a constant prey refuge. We apply optimal control theory to investigate optimal strategies for controlling the threats in the system where anti-poaching patrols are used for controlling poaching, construction of dams for mitigating drought and vaccination for diseases control. The possible impact of using combinations of three controls either one at a time or two at a time on the threatened system plus a refuge factor is examined. All control strategies have shown significant increase in prey and predator populations . However, the best result is achieved by controlling all threats together. The effect of variation of prey refuge   to the control of threats is studied and results indicate that increase of  causes more prey individuals to be saved and reduces the number of predator individuals saved. This behaviour agrees with theoretical results obtained in co-existence equilibrium point.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Optimal Control of a Threatened Wildebeest-lion Prey-predator System Incorporating a Constant Prey Refuge in the Serengeti Ecosystem
AU  - Thadei Damas Sagamiko
AU  - Nyimvua Shaban
AU  - Cuthbert Leonard Nahonyo
AU  - Oluwole Daniel Makinde
Y1  - 2015/07/17
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150404.18
DO  - 10.11648/j.acm.20150404.18
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 296
EP  - 312
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150404.18
AB  - In this paper a two species prey-predator model is developed in which prey is wildebeest and predator is lion and both are threatened by poaching, drought and diseases.The system is found in the Serengeti ecosystem.The model is constructed based on Holling type II functional response incorporating a constant prey refuge. We apply optimal control theory to investigate optimal strategies for controlling the threats in the system where anti-poaching patrols are used for controlling poaching, construction of dams for mitigating drought and vaccination for diseases control. The possible impact of using combinations of three controls either one at a time or two at a time on the threatened system plus a refuge factor is examined. All control strategies have shown significant increase in prey and predator populations . However, the best result is achieved by controlling all threats together. The effect of variation of prey refuge   to the control of threats is studied and results indicate that increase of  causes more prey individuals to be saved and reduces the number of predator individuals saved. This behaviour agrees with theoretical results obtained in co-existence equilibrium point.
VL  - 4
IS  - 4
ER  -

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