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:

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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	M. Groom, Allee effects limit population viability of an annual plant,, The American Naturalist, 151* (1998), 487. doi: 10.1086/286135. Google Scholar*
[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. Google Scholar*
[31]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,, Springer-Verlag, (1983). Google Scholar
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