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

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.

DOI	10.11648/j.acm.20150404.18
Published in	Applied and Computational Mathematics (Volume 4, Issue 4, August 2015)
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), 2024. Published by Science Publishing Group

Keywords

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

References

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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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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

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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}
}

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  -

Copy | Download