Interaction between prey and predator species is a complex and non-linear process. Understanding various phenomena in the dynamics of prey-predator systems is vital to both mathematical ecology and conservation biology. Mathematical models on prey-predator systems have been the hot sport providing important information regarding the interactions of prey and predator species in various ecosystems. In this paper, a review of the available mathematical models on prey-predator systems was done. Our aim was to assess their structure, behaviour, available control strategies, population involved and their ability in predicting the future behaviour of the ecosystems. We observed diversities in the reviewed mathematical models, some model incorporated factors such as drought, harvesting and prey refuge as the factors that affect ecosystems, some ignored the contribution of environmental variations while others considered the variable carrying capacity. Most of the models reviewed have not considered the contribution of diseases and seasonal weather variation in the dynamics of prey predator systems. Some of the reviewed models do not match the real situation in most modelled ecosystems. Thus, to avoid unreliable results, this review reveals the need to incorporate seasonal weather variations and diseases in the dynamics of prey predator systems of Serengeti ecosystem.
References
[1]
Jawad, S. (2018) Modelling, Dynamics and Analysis of Multi-Species Systems with Prey Refuge. PhD Thesis, Brunel University, London.
[2]
Kideghesho, J.R. (2010) Serengeti Shall Not Die: Transforming an Ambition into a Reality. Tropical Conservation Science, 3, 228-247. https://doi.org/10.1177/194008291000300301
[3]
Holling, C.S. (1965) The Functional Response of Predators to Prey Density and Its Role in Mimicry and Population Regulation. The Memoirs of the Entomological Society of Canada, 97, 5-60. https://doi.org/10.4039/entm9745fv
[4]
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. https://doi.org/10.3354/esr00120
[5]
Packer, C., Scheel, D. and Pusey, A.E. (1990) Why Lions form Groups: Food Is Not Enough. The American Naturalist, 136, 1-19. https://doi.org/10.1086/285079
[6]
Sagamiko, T.D., Shaban, N., Nahonyo, C.L. and Makinde, O.D. (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, 296-312. https://doi.org/10.11648/j.acm.20150404.18
[7]
Mduma, S.A.R. (1996) Serengeti Wildebeest Population Dynamics: Regulation, Limitation and Implications for Harvesting. PhD Thesis, The University of British Columbia, Vancouver.
[8]
Fay, T.H. and Greeff, J.C. (2006) Lion, Wildebeest and Zebra: A Predator-Prey Model. Ecological Modelling, 196, 237-244. https://doi.org/10.1016/j.ecolmodel.2006.02.026
[9]
Edwin, A. (2010) Modeling and Analysis of a Two Prey-One Predator System with Harvesting, Holling Type II and Ratio-Dependent Responses. PhD Thesis, Makerere University, Kampala.
[10]
Kideghesho, J.R., Nyahongo, J.W., Hassan, S.N., Tarimo, T.C., Mbije, N.E., et al. (2006) Factors and Ecological Impacts of Wildlife Habitat Destruction in the Serengeti Ecosystem in Northern Tanzania. African Journal of Environmental Assessment and Management, 11, 17-32.
[11]
Sadiq, A. (2017) The Dynamics and Optimal Control of a Prey-Predator System. Global Journal of Pure and Applied Mathematics, 13, 5287-5298.
[12]
Raymond, C., Hugo, A. and Kung-aro, M. (2019) Modeling Dynamics of Prey-Predator Fishery Model with Harvesting: A Bioeconomic Model. Journal of Applied Mathematics, 2019, Article ID: 2601648. https://doi.org/10.1155/2019/2601648
[13]
Wolanski, E. and Gereta, E. (2001) Water Quantity and Quality as the Factors Driving the Serengeti Ecosystem, Tanzania. Hydrobiologia, 458, 169-180. https://doi.org/10.1023/A:1013125321838
[14]
Agarwal, M. and Pathak, R. (2014) Influence of Prey Reserve in Two Preys and One Predator System. International Journal of Engineering, Science and Technology, 6, 1-19. https://doi.org/10.4314/ijest.v6i2.1
[15]
Schaller, G.B. (2009) The Serengeti Lion: A Study of Predator-Prey Relations. University of Chicago Press, Chicago.
[16]
Chakraborty, K., Chakraborty, M. and Kar, T.K. (2011) Optimal Control of Harvest and Bifurcation of a Prey-Predator Model with Stage Structure. Applied Mathematics and Computation, 217, 8778-8792. https://doi.org/10.1016/j.amc.2011.03.139
[17]
Lotka, A.J. (1925) Elements of Physical Biology. Williams and Wilkins, Philadelphia.
[18]
Volterra, V. (1926) Fluctuations in the Abundance of a Species Considered Mathematically. Nature, 118, 558-560. https://doi.org/10.1038/118558a0
[19]
Ngana, J.J., Luboobi, L.S. and Kuznetsov, D. (2014) Modelling the Migratory Population Dynamics of the Serengeti Ecosystem. Applied and Computational Mathematics, 3, 125-129. https://doi.org/10.11648/j.acm.20140304.13
[20]
Hering, R.H. (1990) Oscillations in Lotka-Volterra Systems of Chemical Reactions. Journal of Mathematical Chemistry, 5, 197-202. https://doi.org/10.1007/BF01166429
[21]
Laval, G., Donald Joseph, X.M., Bekefi, G., Bers, A., Pellat, R., Delcroix, J.-L. and Kalman, G. (1975) Plasma Physics. Gordon and Breach Science Publishers, New York.
[22]
Busse, F.H. (1981) Transition to Turbulence via the Statistical Limit Cycle Route. In: Haken, H., Ed., Chaos and Order in Nature, Springer, Berlin, 36-44. https://doi.org/10.1007/978-3-642-68304-6_4
[23]
Solomon, S. and Richmond, P. (2002) Stable Power Laws in Variable Economies; Lotkavolterra Implies Pareto-Zipf. The European Physical Journal B—Condensed Matter and Complex Systems, 27, 257-261. https://doi.org/10.1140/epjb/e20020152
[24]
Bezabih, A.F., Edessa, G.K. and Koya, P.R. (2020) Mathematical Eco-Epidemiological Model on Prey-Predator System. Mathematical Modeling and Applications, 5, 183-190. https://doi.org/10.11648/j.mma.20200503.17
[25]
Gretchko, S., Marley, J. and Tyson, R.C. (2018) The Effects of Climate Change on Predator-Prey Dynamics.
[26]
Bezabih, A.F., Edessa, G.K. and Rao, K.P. (2021) Ecoepidemiological Model and Analysis of Prey-Predator System. Journal of Applied Mathematics, 2021, Article ID: 6679686. https://doi.org/10.1155/2021/6679686
[27]
Savitri, D. (2019) Stability and Numerical Simulation of Prey-Predator System with Holling Type-II Functional Responses for Adult Prey. Journal of Physics: Conference Series, 1417, Article ID: 012025. https://doi.org/10.1088/1742-6596/1417/1/012025
[28]
Sharma, S. and Samanta, G. (2015) Analysis of a Two Prey One Predator System with Disease in the First Prey Population. International Journal of Dynamics and Control, 3, 210-224. https://doi.org/10.1007/s40435-014-0107-4
[29]
Tyson, R. and Lutscher, F. (2016) Seasonally Varying Predation Behavior and Climate Shifts Are Predicted to Affect Predator-Prey Cycles. The American Naturalist, 188, 539-553. https://doi.org/10.1086/688665
[30]
Sauve, A.M., Taylor, R.A. and Barraquand, F. (2020) The Effect of Seasonal Strength and Abruptness on Predator-Prey Dynamics. Journal of Theoretical Biology, 491, Article ID: 110175. https://doi.org/10.1016/j.jtbi.2020.110175
[31]
Greenhalgh, D., Khan, Q.J. and Pettigrew, J.S. (2017) An Eco-Epidemiological Predator-Prey Model Where Predators Distinguish between Susceptible and Infected Prey. Mathematical Methods in the Applied Sciences, 40, 146-166. https://doi.org/10.1002/mma.3974
[32]
Greenhalgh, D., Khan, Q.J. and Al-Kharousi, F.A. (2020) Eco-Epidemiological Model with Fatal Disease in the Prey. Nonlinear Analysis: Real World Applications, 53, Article ID: 103072. https://doi.org/10.1016/j.nonrwa.2019.103072
[33]
Chakraborty, K., Das, K. and Kar, T. (2013) An Ecological Perspective on Marine Reserves in Prey-Predator Dynamics. Journal of Biological Physics, 39, 749-776. https://doi.org/10.1007/s10867-013-9329-5
[34]
Kar, T. and Batabyal, A. (2010) Persistence and Stability of a Two Prey One Predator System. International Journal of Engineering, Science and Technology, 2, 174-190. https://doi.org/10.4314/ijest.v2i2.59164
[35]
Amalia, R.D., Arif, D.K., et al. (2018) Optimal Control of Predator-Prey Mathematical Model with Infection and Harvesting on Prey. Journal of Physics: Conference Series, 974, Article ID: 012050. https://doi.org/10.1088/1742-6596/974/1/012050
[36]
Amalia, A.P., Aldila, D. and Dumbela, P.A. (2020) Eco-Epidemiological Model in the Interaction between Pelecanidae and Tilapia. AIP Conference Proceedings, 2296, Article ID: 020089. https://doi.org/10.1063/5.0030424
[37]
Ganguli, C., Kar, T. and Mondal, P. (2017) Optimal Harvesting of a Prey-Predator Model with Variable Carrying Capacity. International Journal of Biomathematics, 10, Article ID: 1750069. https://doi.org/10.1142/S1793524517500693
[38]
Kar, T. and Chaudhuri, K. (2004) Harvesting in a Two-Prey One-Predator Fishery: A Bioeconomic Model. The ANZIAM Journal, 45, 443-456. https://doi.org/10.1017/S144618110001347X
[39]
Kabuye, R. (1995) Mathematical Analysis of Interaction within a Four Species Ecosystem. Department of Mathematics, Makerere University, Kampala.
[40]
Ashine, A.B. and Gebru, D.M. (2017) Mathematical Modeling of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes. Mathematics and Decision Sciences, 17, 20-40.
[41]
Bennett, V.J. (2003) Computer Modelling the Serengeti-Mara Ecosystem. PhD Thesis, University of Leeds, Leeds.
[42]
Sinclair, A. and Arcese, P. (1995) Population Consequences of Predation-Sensitive Foraging: The Serengeti Wildebeest. Ecology, 76, 882-891. https://doi.org/10.2307/1939353
Pennycuick, C.J. (2008) Modelling the Flying Bird. Elsevier, Amsterdam.
[45]
Sinclair, A.R.E. and Norton-Griffiths, M. (1984) Serengeti: Dynamics of an Ecosystem. University of Chicago Press, Chicago.
[46]
Hopcraft, J.G.C., Holdo, R., Mwangomo, E., Mduma, S., Thirgood, S., Borner, M., Fryxell, J., Olff, H. and Sinclair, A. (2015) Why Are Wildebeest the Most Abundant Herbivore in the Serengeti Ecosystem. In: Serengeti IV: Sustaining Biodiversity in a Coupled Human-Natural System, The University of Chicago Press, Chicago, 125-174.
[47]
Knapp, E.J. (2012) Why Poaching Pays: A Summary of Risks and Benefits Illegal Hunters Face in Western Serengeti, Tanzania. Tropical Conservation Science, 5, 434-445. https://doi.org/10.1177/194008291200500403
[48]
Laws, A.N. (2017) Climate Change Effects on Predator-Prey Interactions. Current Opinion in Insect Science, 23, 28-34. https://doi.org/10.1016/j.cois.2017.06.010
[49]
Mills, M. and Shenk, T. (1992) Predator-Prey Relationships: The Impact of Lion Predation on Wildebeest and Zebra Populations. Journal of Animal Ecology, 61, 693-702. https://doi.org/10.2307/5624
[50]
Safuan, H.M. (2015) Mathematical Analysis of Population Growth Subject to Environmental Change. Bulletin of the Australian Mathematical Society, 92, 351-352. https://doi.org/10.1017/S0004972715000659
[51]
Agnihotri, K.B. and Gakkhar, S. (2012) The Dynamics of Disease Transmission in a Prey Predator System with Harvesting of Prey. International Journal of Advanced Research in Computer Engineering & Technology, 1, 1-17.
[52]
Branagan, D. and Hammond, J. (1965) Rinderpest in Tanganyika: A Review. Bulletin of Epizootic Diseases of Africa, 13, 225-246.
[53]
Hilborn, R. and Sinclair, A. (1979) A Simulation of the Wildebeest Population, Other Ungulates, and Their Predators. In: Sinclair, A.R.E. and Norton-Griffiths, M., Eds., Serengeti: Dynamics of an Ecosystem, Chicago University Press, Chicago, 287-309.
[54]
Sahoo, B. and Poria, S. (2014) Diseased Prey Predator Model with General Holling Type Interactions. Applied Mathematics and Computation, 226, 83-100. https://doi.org/10.1016/j.amc.2013.10.013
[55]
Lolika, P.O. and Mushayabasa, S. (2018) Dynamics and Stability Analysis of a Brucellosis Model with Two Discrete Delays. Discrete Dynamics in Nature and Society, 2018, Article ID: 6456107. https://doi.org/10.1155/2018/6456107
[56]
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. https://doi.org/10.5897/JENE12.013
[57]
Bornaa, C.S., Makinde, O.D. and Seini, I.Y. (2015) Eco-Epidemiological Model and Optimal Control of Disease Transmission between Humans and Animals. Communications in Mathematical Biology and Neuroscience, 2015, Article No. 26.
[58]
Bodine, E.N. (2010) Optimal Control of Species Augmentation Conservation Strategies. Dissertation, University of Tennessee, Knoxville.
[59]
Tayeh, R.L. and Naji, R.K. (2014) Mathematical Study of Eco-Epidemiological System. Mathematical Theory and Modeling, 4, 172-200.
