Dynamics of a predator-prey system with prey subject to Allee effects and disease

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2014, 11(4): 877-918. doi: 10.3934/mbe.2014.11.877

Dynamics of a predator-prey system with prey subject to Allee effects and disease

Yun Kang ^1, , Sourav Kumar Sasmal ^2, , Amiya Ranjan Bhowmick ^2, and Joydev Chattopadhyay ^2,

1.	Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212
2.	Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India, India, India

Received March 2013 Revised September 2013 Published March 2014

Abstract
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In this article, we propose a general predator-prey system where prey is subject to Allee effects and disease with the following unique features: (i) Allee effects built in the reproduction process of prey where infected prey (I-class) has no contribution; (ii) Consuming infected prey would contribute less or negatively to the growth rate of predator (P-class) in comparison to the consumption of susceptible prey (S-class). We provide basic dynamical properties for this general model and perform the detailed analysis on a concrete model (SIP-Allee Model) as well as its corresponding model in the absence of Allee effects (SIP-no-Allee Model); we obtain the complete dynamics of both models: (a) SIP-Allee Model may have only one attractor (extinction of all species), two attractors (bi-stability either induced by small values of reproduction number of both disease and predator or induced by competition exclusion), or three attractors (tri-stability); (b) SIP-no-Allee Model may have either one attractor (only S-class survives or the persistence of S and I-class or the persistence of S and P-class) or two attractors (bi-stability with the persistence of S and I-class or the persistence of S and P-class). One of the most interesting findings is that neither models can support the coexistence of all three S, I, P-class. This is caused by the assumption (ii), whose biological implications are that I and P-class are at exploitative competition for S-class whereas I-class cannot be superior and P-class cannot gain significantly from its consumption of I-class. In addition, the comparison study between the dynamics of SIP-Allee Model and SIP-no-Allee Model lead to the following conclusions: 1) In the presence of Allee effects, species are prone to extinction and initial condition plays an important role on the surviving of prey as well as its corresponding predator; 2) In the presence of Allee effects, disease may be able to save prey from the predation-driven extinction and leads to the coexistence of S and I-class while predator can not save the disease-driven extinction. All these findings may have potential applications in conservation biology.

Keywords: eco-epidemiological system., bi-stability, functional responses, disease/predation-driven extinction, tri-stability, Allee effect.

Mathematics Subject Classification: Primary: 35B32, 37C75; Secondary: 92B05, 92D30, 92D4.

Citation: Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877

References:

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References:

[1]	W. C. Allee, Animal Aggregations. A Study in General Sociology,, University of Chicago Press, (1931). doi: 10.5962/bhl.title.7313.
[2]	L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation,, Mathematical Biosciences, 152* (1998), 63. doi: 10.1016/S0025-5564(98)10018-4.*
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[8]	E. Beretta and Y. Kuang, Modelling and analysis of a marine bacteriophage infection,, Mathematical Biosciences, 149* (1998), 57. doi: 10.1016/S0025-5564(97)10015-3.*
[9]	F. S. Berezovskaya, B. Song and C. Castillo-Chavez, Role of prey dispersal and refuges on predator-prey dynamics,, SIAM Journal on Applied Mathematics, 70 (2010), 1821. doi: 10.1137/080730603.
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[11]	D. S. Boukal and L. Berec, Single-species Models of the Allee effect: Extinction boundaries, sex ratios and mate encounters,, Journal of Theoretical Biology, 218* (2002), 375. doi: 10.1006/jtbi.2002.3084.*
[12]	R. Burrows, H. Hofer and M. L. East, Population dynamics, intervention and survival in African wild dogs (Lycaon pictus),, Proceedings of the Royal Society B: Biological Sciences, 262* (1995), 235. doi: 10.1098/rspb.1995.0201.*
[13]	J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey,, Nonlinear Analysis, 36 (1999), 747. doi: 10.1016/S0362-546X(98)00126-6.
[14]	J. Chattopadhyay and S. Pal, Viral infection on phytoplankton-zooplankton system-a mathematical model,, Ecological Modelling, 151* (2002), 15. doi: 10.1016/S0304-3800(01)00415-X.*
[15]	J. Chattopadhyay, R. Sarkar, M. E. Fritzche-Hoballah, T. Turlings and L. Bersier, Parasitoids may determine plant fitness - A mathematical model based on experimental data,, Journal of Theoretical Biology, 212* (2001), 295. doi: 10.1006/jtbi.2001.2374.*
[16]	J. Chattopadhyay, P. Srinivasu and N. Bairagi, Pelicans at risk in Salton Sea-an eco-epidemiological model-II,, Ecological Modelling, 167* (2003), 199. doi: 10.1016/S0304-3800(03)00187-X.*
[17]	D. L. Clifford, J. A. K. Mazet, E. J. Dubovi, D. K. Garcelon, T. J. Coonan, P. A. Conrad and L. Munson, Pathogen exposure in endangered island fox (Urocyon littoralis) populations: Implications for conservation management,, Biological Conservation, 131* (2006), 230. doi: 10.1016/j.biocon.2006.04.029.*
[18]	F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001.
[19]	F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect,, Trends in Ecology & Evolution, 14 (1999), 405. doi: 10.1016/S0169-5347(99)01683-3.
[20]	F. Courchamp, T. Clutton-Brock and B. Grenfell, Multipack dynamics and the Allee effect in the African wild dog, Lycaon pictus,, Animal Conservation, 3 (2000), 277. doi: 10.1017/S1367943000001001.
[21]	F. Courchamp, B. Grenfell and T. Clutton-Brock, Impact of natural enemies on obligately cooperatively breeders,, Oikos, 91 (2000), 311. doi: 10.1034/j.1600-0706.2000.910212.x.
[22]	J. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects,, Journal of Biological Dynamics, 6 (2012), 941. doi: 10.1080/17513758.2012.697196.
[23]	A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism,, Oikos, 112* (2006), 667. doi: 10.1111/j.0030-1299.2006.14243.x.*
[24]	J. Drake, Allee effects and the risk of biological invasion,, Risk Analysis, 24 (2004), 795. doi: 10.1111/j.0272-4332.2004.00479.x.
[25]	J. Ferdy, F. Austerlitz, J. Moret, P. Gouyon and B. Godelle, Pollinator-induced density dependence in deceptive species,, Oikos, 87 (1999), 549. doi: 10.2307/3546819.
[26]	H. I. Freedman, A model of predator-prey dynamics as modified by the action of parasite,, Mathematical Biosciences, 99 (1990), 143. doi: 10.1016/0025-5564(90)90001-F.
[27]	A. Friedman and A. A. Yakubu, Fatal disease and demographic allee effect: Population persistence and extinction,, Journal of Biological Dynamics, 6 (2012), 495. doi: 10.1080/17513758.2011.630489.
[28]	J. C. Gascoigne and R. N. Lipccius, Allee effects driven by predation,, Journal of Applied Ecology, 41 (2004), 801. doi: 10.1111/j.0021-8901.2004.00944.x.
[29]	M. Groom, Allee effects limit population viability of an annual plant,, The American Naturalist, 151* (1998), 487. doi: 10.1086/286135.*
[30]	Y. Gruntfest, R. Arditi and Y. Dombronsky, A fragmented population in a varying environment,, Journal of Theoretical Biology, 185* (1997), 539. doi: 10.1006/jtbi.1996.0358.*
[31]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,, Springer-Verlag, (1983).
[32]	F. M. D. Gulland, The Impact of Infectious Diseases on Wild Animal Populations-A Review. In: Ecology of Infectious Diseases in Natural Populations,, Cambridge University Press, (1995).
[33]	K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection,, Journal of Mathematical Biology, 27 (1989), 609. doi: 10.1007/BF00276947.
[34]	H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 499. doi: 10.1137/S0036144500371907.
[35]	H. W. Hethcote, W. Wang, L. Han and Z. Ma, A predator-prey model with infected prey,, Theoretical Population Biology, 66 (2004), 259. doi: 10.1016/j.tpb.2004.06.010.
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