Modelling and analysis of a modified May-Holling-Tanner predator-prey model with Allee effect in the prey and an alternative food source for the predator

Claudio Arancibia–Ibarra; José Flores; Claudio Arancibia–Ibarra; José Flores

doi:10.3934/mbe.2020408

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 8052-8073. doi: 10.3934/mbe.2020408

Previous Article Next Article

Research article Special Issues

Modelling and analysis of a modified May-Holling-Tanner predator-prey model with Allee effect in the prey and an alternative food source for the predator

Claudio Arancibia–Ibarra ^{1,2
,
,},
José Flores ³

1.
School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
2.
Facultad de Ingeniería y Negocios, Universidad de Las Américas, Santiago, Chile
3.
Department of Mathematics, The University of South Dakota, Vermillion, South Dakota, USA

Received: 24 June 2020 Accepted: 22 October 2020 Published: 12 November 2020

In the present study, we have modified the traditional May-Holling-Tanner predator-prey model used to represent the interaction between least-weasel and field-vole population by adding an Allee effect (strong and weak) on the field-vole population and alternative food source for the weasel population. It is shown that the dynamic is different from the original May-Holling-Tanner predator-prey interaction since new equilibrium points have appeared in the first quadrant. Moreover, the modified model allows the extinction of both species when the Allee effect (strong and weak) on the prey is included, while the inclusion of the alternative food source for the predator shows that the system can support the coexistence of the populations, extinction of the prey and coexistence and oscillation of the populations at the same time. Furthermore, we use numerical simulations to illustrate the impact that changing the predation rate and the predator intrinsic growth rate have on the basin of attraction of the stable equilibrium point or stable limit cycle in the first quadrant. These simulations show the stabilisation of predator and prey populations and/or the oscillation of these two species over time.
- modified May-Holling-Tanner,
- Holling type II,
- alternative food,
- Allee effect
Citation: Claudio Arancibia–Ibarra, José Flores. Modelling and analysis of a modified May-Holling-Tanner predator-prey model with Allee effect in the prey and an alternative food source for the predator[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 8052-8073. doi: 10.3934/mbe.2020408

Related Papers:

Abstract

In the present study, we have modified the traditional May-Holling-Tanner predator-prey model used to represent the interaction between least-weasel and field-vole population by adding an Allee effect (strong and weak) on the field-vole population and alternative food source for the weasel population. It is shown that the dynamic is different from the original May-Holling-Tanner predator-prey interaction since new equilibrium points have appeared in the first quadrant. Moreover, the modified model allows the extinction of both species when the Allee effect (strong and weak) on the prey is included, while the inclusion of the alternative food source for the predator shows that the system can support the coexistence of the populations, extinction of the prey and coexistence and oscillation of the populations at the same time. Furthermore, we use numerical simulations to illustrate the impact that changing the predation rate and the predator intrinsic growth rate have on the basin of attraction of the stable equilibrium point or stable limit cycle in the first quadrant. These simulations show the stabilisation of predator and prey populations and/or the oscillation of these two species over time.

References

[1]	I. Hanski, L. Hansson, H. Henttonen, Specialist predators, generalist predators, and the microtine rodent cycle, J. Anim. Ecol., 60 (1991), 353-367. doi: 10.2307/5465
[2]	I. Hanski, H. Henttonen, E. Korpimäki, L. Oksanen, P. Turchin, Small-rodent dynamics and predation, Ecology, 82 (2001), 15005-1520.
[3]	P. Turchin, I. Hanski, An empirically based model for latitudinal gradient in vole population dynamics, Am. Nat., 149 (1997), 842-874. doi: 10.1086/286027
[4]	T. Dalkvist, R. Sibly, C. Topping, How predation and landscape fragmentation affect vole population dynamics, Plos One, 149 (2011), 1-8.
[5]	D. Madison, R. FitzGerald, W. McShea, Dynamics of social nesting in overwintering meadow voles (Microtus pennsylvanicus): possible consequences for population cycling, Behav. Ecol. Sociobiol., 15 (1984), 9-17. doi: 10.1007/BF00310209
[6]	M. Graham, X. Lambin, The impact of weasel predation on cyclic field-vole survival: the specialist predator hypothesis contradicted, J. Anim. Ecol., 71 (2002), 946-956. doi: 10.1046/j.1365-2656.2002.00657.x
[7]	P. Turchin, Complex population dynamics: a theoretical/empirical synthesis, Monographs in population biology, Princeton University Press, Princeton N.J., 2003.
[8]	R. May, Stability and complexity in model ecosystems, Monographs in population biology, Princeton University Press, Princeton N.J., 1974.
[9]	p. Leslie, J. Gower, The Properties of a Stochastic Model for the Predator-Prey Type of Interaction Between Two Species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219
[10]	E. Pielou, An introduction to Mathematical Ecology, Wiley-Interscience, New York, 1974.
[11]	X. Santos, M. Cheylan, Taxonomic and functional response of a Mediterranean reptile assemblage to a repeated fire regime, Biol. Conserv., 168 (2013), 90-98. doi: 10.1016/j.biocon.2013.09.008
[12]	P. Aguirre, E. González-Olivares, E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Anal.-Real., 10 (2009), 1401-1416.
[13]	C. Arancibia-Ibarra, The basins of attraction in a Modified May-Holling-Tanner predator-prey model with Allee effect, Nonlinear Anal.-Theor., 185 (2019), 15-28. doi: 10.1016/j.na.2019.03.004
[14]	E. Sáez, E. González-Olivares, Dynamics on a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.
[15]	F. Dumortier, J. Llibre, J. C. Artés, Qualitative theory of planar differential systems, Springer Berlin Heidelberg, Springer-Verlag Berlin Heidelberg, 2006.
[16]	B. Lisena, Global stability of a periodic Holling-Tanner predator-prey model, Math. Method. Appl. Sci., 41 (2018), 3270-3281. doi: 10.1002/mma.4814
[17]	H. Cao, Z. Yue, Y. Zhou, The stability and bifurcation analysis of a discrete Holling-Tanner model, Adv. Differ. Equ., 2013 (2013), 1-17. doi: 10.1186/1687-1847-2013-1
[18]	W. Allee, The social life of animals, WW Norton & Co, New York, 1938.
[19]	W. Allee, O. Park, A. Emerson, T. Park, K. Schmidt, Principles of animal ecology, WB Saundere Co. Ltd., Philadelphia, 1949.
[20]	L. Berec, E. Angulo, F. Courchamp, Multiple Allee effects and population management, Trends. Ecol. Evol., 22 (2007), 185-191. doi: 10.1016/j.tree.2006.12.002
[21]	G. Buffoni, M. Groppi, C. Soresina, Dynamics of predator-prey models with a strong Allee effect on the prey and predator-dependent trophic functions, Nonlinear Anal.-Real., 30 (2016), 143-169. doi: 10.1016/j.nonrwa.2015.12.001
[22]	G. van Voorn, L. Hemerik, M. Boer, B. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469. doi: 10.1016/j.mbs.2007.02.006
[23]	J. Wang, J. Shi, J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331. doi: 10.1007/s00285-010-0332-1
[24]	article P. Stephens, W. Sutherland, R. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190. doi: 10.2307/3547011
[25]	M. Jankovic, S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z
[26]	D. Morris, Measuring the Allee effect: positive density dependence in small mammals, Ecology, 83 (2002), 14-20. doi: 10.1890/0012-9658(2002)083[0014:MTAEPD]2.0.CO;2
[27]	R. Ostfeld, C. Canham, Density-dependent processes in meadow voles: an experimental approach, Ecology, 76 (1995), 521-532. doi: 10.2307/1941210
[28]	F. Courchamp, L. Berec, J. Gascoigne, Allee effects in ecology and conservation, Oxford University Press, 2008.
[29]	M. Liermann, R. Hilborn, Depensation: evidence, models and implications, Fish. Fish., 2 (2001), 33-58, doi: 10.1046/j.1467-2979.2001.00029.x
[30]	R. McDonald, C. Webbon, S. Harris, The diet of stoats (Mustela erminea) and weasels (Mustela nivalis) in Great Britain, J. Zool., 252 (2000), 363-371. doi: 10.1111/j.1469-7998.2000.tb00631.x
[31]	M. Andersson, S. Erlinge, Influence of predation on rodent populations, Oikos, 29 (1977), 591-597. doi: 10.2307/3543597
[32]	S. Erlinge, Predation and noncyclicity in a microtine population in southern Sweden, Oikos, 50 (1987), 347-352. doi: 10.2307/3565495
[33]	I. Hanski, P. Turchin, E. Korpimaki, H. Henttonen, Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos, Nature, 364 (1993), 232-235. doi: 10.1038/364232a0
[34]	L. Hansson, Competition between rodents in successional stages of taiga forests: Microtus agrestis vs. Clethrionomys glareolus, Oikos, 40 (1983), 258-266.
[35]	D. Wollkind, J. Collings, J. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, B. Math. Biol., 50 (1988), 379-409. doi: 10.1016/S0092-8240(88)90005-5
[36]	M. Aziz-Alaoui, M. Daher, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6
[37]	A. Dhooge, W. Govaerts, V. Kuznetsov, H. Meijer, B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comp. Model. Dyn., 14 (2008), 147-175. doi: 10.1080/13873950701742754
[38]	J. Huang, Y. Gong, S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete. Cont. Dyn-B., 18 (2013), 2101-2121.
[39]	P. Roux, J. Shaw, S. Chown, Ontogenetic shifts in plant interactions vary with environmental severity and affect population structure, New. Phytol., 200 (2013), 241-250. doi: 10.1111/nph.12349
[40]	C. Arancibia-Ibarra, J. Flores, G. Pettet, P. van Heijster, A Holling-Tanner predator-prey model with strong Allee effect, Int. J. Bifurcat. Chaos, 29 (2019), 1-16.
[41]	C. Arancibia-Ibarra, E. González-Olivares, The Holling-Tanner model considering an alternative food for predator, Proc. CMMSE 2015, (2015), 130-141.
[42]	E. González-Olivares, C. Arancibia-Ibarra, A. Rojas-Palma, B. González-Ya?ez, Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators, Math. Biosci. Eng., 16 (2019), 4274-4298.
[43]	C. Arancibia-Ibarra, E. González-Olivares, A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey, BIOMAT Int. S. Math. Co., 1(2011), 46-162.
[44]	E. González-Olivares, C. Arancibia-Ibarra, A. Rojas-Palma, B. González-Ya?ez, Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response, Math. Biosci. Eng., 16 (2019), 7995-8024.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)