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

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

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.
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:
[1]

W. C. Allee, Animal Aggregations. A Study in General Sociology,, University of Chicago Press, (1931). doi: 10.5962/bhl.title.7313. Google Scholar

[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. Google Scholar

[3]

P. Amarasekare, Interactions between local dynamics and dispersal: Insights from single species models,, Theoretical Population Biology, 53 (1998), 44. doi: 10.1006/tpbi.1997.1340. Google Scholar

[4]

E. Angulo, G. W. Roemer, L. Berec, J. Gascoigen and F. Courchamp, Double Allee effects and extinction in the island fox,, Conservation Biology, 21 (2007), 1082. doi: 10.1111/j.1523-1739.2007.00721.x. Google Scholar

[5]

N. Bairagi, P. K. Roy and J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functions - A comparative study,, Journal of Theoretical Biology, 248 (2007), 10. doi: 10.1016/j.jtbi.2007.05.005. Google Scholar

[6]

M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: Numbers, densities and areas,, Epidemiology and Infection, 129 (2002), 147. doi: 10.1017/S0950268802007148. Google Scholar

[7]

E. Beltrami and T. O. Carroll, Modelling the role of viral disease in recurrent phytoplankton blooms,, Journal of Mathematical Biology, 32 (1994), 857. doi: 10.1007/BF00168802. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations,, Massachusetts, (1982). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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

[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). Google Scholar

[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. Google Scholar

[34]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 499. doi: 10.1137/S0036144500371907. Google Scholar

[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. Google Scholar

[36]

F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect,, Journal of Biological Dynamics, 4 (2010), 86. doi: 10.1080/17513750903026429. Google Scholar

[37]

F. M. Hilker, M. Langlais and H. Malchow, The allee effect and infectious diseases: Extinction, multistability, and the (dis-)appearance of oscillations,, The American Naturalist, 173 (2009), 72. doi: 10.1086/593357. Google Scholar

[38]

F. M. Hilker, M. Langlais, S. V. Petrovskii and H. Malchow, A diffusive SI model with Allee effect and application to FIV,, Mathematical Biosciences, 206 (2007), 61. doi: 10.1016/j.mbs.2005.10.003. Google Scholar

[39]

F. M. Hilker and K. Schmitz, Disease-induced stabilization of predator-prey oscillations,, Journal of Theoretical Biology, 255 (2008), 299. doi: 10.1016/j.jtbi.2008.08.018. Google Scholar

[40]

C. Holling, Some characteristics of simple types of predation and parasitism,, Canadian Entomologist, 91 (1959), 385. Google Scholar

[41]

V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems,, Mathematical Biosciences, 111 (1992), 1. doi: 10.1016/0025-5564(92)90078-B. Google Scholar

[42]

J. Jacobs, Cooperation, optimal density and low density thresholds: yet another modification of the logistic model,, Oecologia, 64 (1984), 389. doi: 10.1007/BF00379138. Google Scholar

[43]

S. R. J. Jang, Discrete-time host-parasitoid models with Allee effects: Density dependence versus parasitism,, Journal of Difference Equations and Applications, 17 (2011), 525. doi: 10.1080/10236190903146920. Google Scholar

[44]

S. R. J. Jang, Allee effects in a discrete-time host-parasitoid model,, Journal of Difference Equations and Applications, 12 (2006), 165. doi: 10.1080/10236190500539238. Google Scholar

[45]

Y. Kang, Dynamics of A General Contest Competition Two Species Model Subject to Strong Allee Effects,, Submitted to the Journal of Theoretical Population Biology, (2013). Google Scholar

[46]

Y. Kang, Scramble competitions can rescue endangered species subject to strong Allee effects,, Mathematical Biosciences, 241 (2013), 75. doi: 10.1016/j.mbs.2012.09.002. Google Scholar

[47]

Y. Kang, A. R. Bhowmick, S. K. Sasmal and J. Chattopadhyay, Host-parasitoid systems with predation-driven Allee effects in host population,, preprint., (). Google Scholar

[48]

Y. Kang and C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal,, Journal of Biological Dynamics, 6 (2012), 50. doi: 10.1080/17513758.2012.713125. Google Scholar

[49]

Y. Kang and C. Castillo-Chavez, A Simple Epidemiological Model for Populations in The Wild with Allee Effects and Disease Modified Fitness,, Journal of Discrete and Continuous Dynamical Systems-B, (2013). Google Scholar

[50]

Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects,, Journal of Mathematical Biology, 62 (2011), 925. doi: 10.1007/s00285-010-0359-3. Google Scholar

[51]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator,, Journal of Mathematical Biology, 67 (2013), 1227. doi: 10.1007/s00285-012-0584-z. Google Scholar

[52]

Y. Kang and A.-A. Yakubu, Weak Allee effects and species coexistence,, Nonlinear Analysis: Real World Applications, 12 (2011), 3329. doi: 10.1016/j.nonrwa.2011.05.031. Google Scholar

[53]

M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly,, Oikos, 82 (1998), 384. doi: 10.2307/3546980. Google Scholar

[54]

B. Lamont, P. Klinkhamer and E. Witkowski, Population fragmentation may reduce fertility to zero in Banksia goodii-demonstration of the Allee effect,, Oecologia, 94 (1993), 446. doi: 10.1007/BF00317122. Google Scholar

[55]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theoretical Population Biology, 43 (1993), 141. doi: 10.1006/tpbi.1993.1007. Google Scholar

[56]

H. McCallum, N. Barlow and J. Hone, How should pathogen transmission be modelled?, Trends in Ecology & Evolution, 16 (2001), 295. doi: 10.1016/S0169-5347(01)02144-9. Google Scholar

[57]

H. T. Odum and W. C. Allee, A note on the stable point of populations showing both intraspecific cooperation and disoperation,, Ecology, 35 (1954), 95. doi: 10.2307/1931412. Google Scholar

[58]

V. Padrón and M. C. Trevisan, Effect of aggregating behavior on population recovery on a set of habitat islands,, Mathematical Biosciences, 165 (2000), 63. doi: 10.1016/S0025-5564(00)00005-5. Google Scholar

[59]

A. Potapov, E. Merrill and M. A. Lewis, Wildlife disease elimination and density dependence,, Proceedings of the Royal Society - Biological Sciences, 279 (2012), 3139. doi: 10.1098/rspb.2012.0520. Google Scholar

[60]

R. Ricklefs and G. Miller, Ecology,, Williams and Wilkins Co., (2000). Google Scholar

[61]

B.-E. Sther, T. Ringsby and E. Rskaft, Life history variation, population processes and priorities in species conservation: towards a reunion of research paradigms,, Oikos, 77 (1996), 217. doi: 10.2307/3546060. Google Scholar

[62]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

[63]

J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect,, Journal of Mathematical Biology, 52 (2006), 807. doi: 10.1007/s00285-006-0373-7. Google Scholar

[64]

M. Sieber and F. M. Hilker, The hydra effect in predator-prey models,, Journal of Mathematical Biology, 64 (2012), 341. doi: 10.1007/s00285-011-0416-6. Google Scholar

[65]

B. K. Singh, J. Chattopadhyay and S. Sinha, The role of virus infection in a simple phytoplankton zooplankton system,, Journal of Theoretical Biology, 231 (2004), 153. doi: 10.1016/j.jtbi.2004.06.010. Google Scholar

[66]

P. Stephens and W. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation,, Trends in Ecology & Evolution, 14 (1999), 401. doi: 10.1016/S0169-5347(99)01684-5. Google Scholar

[67]

P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185. doi: 10.2307/3547011. Google Scholar

[68]

A. Stoner and M. Ray-Culp, Evidence for Allee effects in an over-harvested marine gastropod: Density dependent mating and egg production,, Marine Ecology Progress Series, 202 (2000), 297. doi: 10.3354/meps202297. Google Scholar

[69]

M. Su and C. Hui, An ecoepidemiological system with infected predator,, in 3rd International Conference on Biomedical Engineering and Informatics (BMET 2010), 6 (2010), 2390. doi: 10.1109/BMEI.2010.5639698. Google Scholar

[70]

M. Su, C. Hui, Y. Zhang and Z. Li, Spatiotemporal dynamics of the epidemic transmission in a predator-prey syatem,, Bulletin of Mathematical Biology, 70 (2008), 2195. doi: 10.1007/s11538-008-9340-3. Google Scholar

[71]

C. Taylor and A. Hastings, Allee effects in biological invasions,, Ecology Letters, 8 (2005), 895. doi: 10.1111/j.1461-0248.2005.00787.x. Google Scholar

[72]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, Journal of Mathematical Biology, 30 (1992), 755. doi: 10.1007/BF00173267. Google Scholar

[73]

H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious diseases and Allee effect?, Journal of Biological Dynamics, 3 (2009), 305. doi: 10.1080/17513750802376313. Google Scholar

[74]

E. Venturino, Epidemics in predator-prey models: Disease in the prey,, In Arino, 1 (1995), 381. doi: 10.1093/imammb19.3.185. Google Scholar

[75]

E. Venturino, Epidemics in predator-prey models: Disease in the predators,, IMA Journal of Mathematics Applied in Medicine and Biology, 19 (2002), 185. doi: 10.1093/imammb19.3.185. Google Scholar

[76]

G. A. K. v. Voorn, L. Hemerik, M. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitaion in predator-prey systems with a strong Allee effect,, Mathematical Biosciences, 209 (2007), 451. doi: 10.1016/j.mbs.2007.02.006. Google Scholar

[77]

J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey,, Journal of Mathematical Biology, 62 (2011), 291. doi: 10.1007/s00285-010-0332-1. Google Scholar

[78]

M. Wang, M. Kot and M. Neubert, Integrodifference equations, Allee effects, and invasions,, Journal of Mathematical Biology, 44 (2002), 150. doi: 10.1007/s002850100116. Google Scholar

[79]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Texts in Applied Mathematics, 2 (1990). doi: 10.1007/978-1-4757-4067-7. Google Scholar

[80]

Y. Xiao and L. Chen, Modelling and analysis of a predator-prey model with disease in the prey,, Mathematical Biosciences, 171 (2001), 59. doi: 10.1016/S0025-5564(01)00049-9. Google Scholar

[81]

A. A. Yakubu, Allee effects in a discrete-time SIS epidemic model with infected newborns,, Journal of Difference Equations and Applications, 13 (2007), 341. doi: 10.1080/10236190601079076. Google Scholar

[82]

S. R. Zhou, C. Z. Liu and G. Wang, The competitive dynamics of metapopulation subject to the Allee-like effect,, Theoretical Population Biology, 65 (2004), 29. doi: 10.1016/j.tpb.2003.08.002. Google Scholar

show all references

References:
[1]

W. C. Allee, Animal Aggregations. A Study in General Sociology,, University of Chicago Press, (1931). doi: 10.5962/bhl.title.7313. Google Scholar

[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. Google Scholar

[3]

P. Amarasekare, Interactions between local dynamics and dispersal: Insights from single species models,, Theoretical Population Biology, 53 (1998), 44. doi: 10.1006/tpbi.1997.1340. Google Scholar

[4]

E. Angulo, G. W. Roemer, L. Berec, J. Gascoigen and F. Courchamp, Double Allee effects and extinction in the island fox,, Conservation Biology, 21 (2007), 1082. doi: 10.1111/j.1523-1739.2007.00721.x. Google Scholar

[5]

N. Bairagi, P. K. Roy and J. Chattopadhyay, Role of infection on the stability of a predator-prey system with several response functions - A comparative study,, Journal of Theoretical Biology, 248 (2007), 10. doi: 10.1016/j.jtbi.2007.05.005. Google Scholar

[6]

M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: Numbers, densities and areas,, Epidemiology and Infection, 129 (2002), 147. doi: 10.1017/S0950268802007148. Google Scholar

[7]

E. Beltrami and T. O. Carroll, Modelling the role of viral disease in recurrent phytoplankton blooms,, Journal of Mathematical Biology, 32 (1994), 857. doi: 10.1007/BF00168802. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations,, Massachusetts, (1982). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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

[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). Google Scholar

[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. Google Scholar

[34]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 499. doi: 10.1137/S0036144500371907. Google Scholar

[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. Google Scholar

[36]

F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect,, Journal of Biological Dynamics, 4 (2010), 86. doi: 10.1080/17513750903026429. Google Scholar

[37]

F. M. Hilker, M. Langlais and H. Malchow, The allee effect and infectious diseases: Extinction, multistability, and the (dis-)appearance of oscillations,, The American Naturalist, 173 (2009), 72. doi: 10.1086/593357. Google Scholar

[38]

F. M. Hilker, M. Langlais, S. V. Petrovskii and H. Malchow, A diffusive SI model with Allee effect and application to FIV,, Mathematical Biosciences, 206 (2007), 61. doi: 10.1016/j.mbs.2005.10.003. Google Scholar

[39]

F. M. Hilker and K. Schmitz, Disease-induced stabilization of predator-prey oscillations,, Journal of Theoretical Biology, 255 (2008), 299. doi: 10.1016/j.jtbi.2008.08.018. Google Scholar

[40]

C. Holling, Some characteristics of simple types of predation and parasitism,, Canadian Entomologist, 91 (1959), 385. Google Scholar

[41]

V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems,, Mathematical Biosciences, 111 (1992), 1. doi: 10.1016/0025-5564(92)90078-B. Google Scholar

[42]

J. Jacobs, Cooperation, optimal density and low density thresholds: yet another modification of the logistic model,, Oecologia, 64 (1984), 389. doi: 10.1007/BF00379138. Google Scholar

[43]

S. R. J. Jang, Discrete-time host-parasitoid models with Allee effects: Density dependence versus parasitism,, Journal of Difference Equations and Applications, 17 (2011), 525. doi: 10.1080/10236190903146920. Google Scholar

[44]

S. R. J. Jang, Allee effects in a discrete-time host-parasitoid model,, Journal of Difference Equations and Applications, 12 (2006), 165. doi: 10.1080/10236190500539238. Google Scholar

[45]

Y. Kang, Dynamics of A General Contest Competition Two Species Model Subject to Strong Allee Effects,, Submitted to the Journal of Theoretical Population Biology, (2013). Google Scholar

[46]

Y. Kang, Scramble competitions can rescue endangered species subject to strong Allee effects,, Mathematical Biosciences, 241 (2013), 75. doi: 10.1016/j.mbs.2012.09.002. Google Scholar

[47]

Y. Kang, A. R. Bhowmick, S. K. Sasmal and J. Chattopadhyay, Host-parasitoid systems with predation-driven Allee effects in host population,, preprint., (). Google Scholar

[48]

Y. Kang and C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal,, Journal of Biological Dynamics, 6 (2012), 50. doi: 10.1080/17513758.2012.713125. Google Scholar

[49]

Y. Kang and C. Castillo-Chavez, A Simple Epidemiological Model for Populations in The Wild with Allee Effects and Disease Modified Fitness,, Journal of Discrete and Continuous Dynamical Systems-B, (2013). Google Scholar

[50]

Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects,, Journal of Mathematical Biology, 62 (2011), 925. doi: 10.1007/s00285-010-0359-3. Google Scholar

[51]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator,, Journal of Mathematical Biology, 67 (2013), 1227. doi: 10.1007/s00285-012-0584-z. Google Scholar

[52]

Y. Kang and A.-A. Yakubu, Weak Allee effects and species coexistence,, Nonlinear Analysis: Real World Applications, 12 (2011), 3329. doi: 10.1016/j.nonrwa.2011.05.031. Google Scholar

[53]

M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly,, Oikos, 82 (1998), 384. doi: 10.2307/3546980. Google Scholar

[54]

B. Lamont, P. Klinkhamer and E. Witkowski, Population fragmentation may reduce fertility to zero in Banksia goodii-demonstration of the Allee effect,, Oecologia, 94 (1993), 446. doi: 10.1007/BF00317122. Google Scholar

[55]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theoretical Population Biology, 43 (1993), 141. doi: 10.1006/tpbi.1993.1007. Google Scholar

[56]

H. McCallum, N. Barlow and J. Hone, How should pathogen transmission be modelled?, Trends in Ecology & Evolution, 16 (2001), 295. doi: 10.1016/S0169-5347(01)02144-9. Google Scholar

[57]

H. T. Odum and W. C. Allee, A note on the stable point of populations showing both intraspecific cooperation and disoperation,, Ecology, 35 (1954), 95. doi: 10.2307/1931412. Google Scholar

[58]

V. Padrón and M. C. Trevisan, Effect of aggregating behavior on population recovery on a set of habitat islands,, Mathematical Biosciences, 165 (2000), 63. doi: 10.1016/S0025-5564(00)00005-5. Google Scholar

[59]

A. Potapov, E. Merrill and M. A. Lewis, Wildlife disease elimination and density dependence,, Proceedings of the Royal Society - Biological Sciences, 279 (2012), 3139. doi: 10.1098/rspb.2012.0520. Google Scholar

[60]

R. Ricklefs and G. Miller, Ecology,, Williams and Wilkins Co., (2000). Google Scholar

[61]

B.-E. Sther, T. Ringsby and E. Rskaft, Life history variation, population processes and priorities in species conservation: towards a reunion of research paradigms,, Oikos, 77 (1996), 217. doi: 10.2307/3546060. Google Scholar

[62]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

[63]

J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect,, Journal of Mathematical Biology, 52 (2006), 807. doi: 10.1007/s00285-006-0373-7. Google Scholar

[64]

M. Sieber and F. M. Hilker, The hydra effect in predator-prey models,, Journal of Mathematical Biology, 64 (2012), 341. doi: 10.1007/s00285-011-0416-6. Google Scholar

[65]

B. K. Singh, J. Chattopadhyay and S. Sinha, The role of virus infection in a simple phytoplankton zooplankton system,, Journal of Theoretical Biology, 231 (2004), 153. doi: 10.1016/j.jtbi.2004.06.010. Google Scholar

[66]

P. Stephens and W. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation,, Trends in Ecology & Evolution, 14 (1999), 401. doi: 10.1016/S0169-5347(99)01684-5. Google Scholar

[67]

P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185. doi: 10.2307/3547011. Google Scholar

[68]

A. Stoner and M. Ray-Culp, Evidence for Allee effects in an over-harvested marine gastropod: Density dependent mating and egg production,, Marine Ecology Progress Series, 202 (2000), 297. doi: 10.3354/meps202297. Google Scholar

[69]

M. Su and C. Hui, An ecoepidemiological system with infected predator,, in 3rd International Conference on Biomedical Engineering and Informatics (BMET 2010), 6 (2010), 2390. doi: 10.1109/BMEI.2010.5639698. Google Scholar

[70]

M. Su, C. Hui, Y. Zhang and Z. Li, Spatiotemporal dynamics of the epidemic transmission in a predator-prey syatem,, Bulletin of Mathematical Biology, 70 (2008), 2195. doi: 10.1007/s11538-008-9340-3. Google Scholar

[71]

C. Taylor and A. Hastings, Allee effects in biological invasions,, Ecology Letters, 8 (2005), 895. doi: 10.1111/j.1461-0248.2005.00787.x. Google Scholar

[72]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, Journal of Mathematical Biology, 30 (1992), 755. doi: 10.1007/BF00173267. Google Scholar

[73]

H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious diseases and Allee effect?, Journal of Biological Dynamics, 3 (2009), 305. doi: 10.1080/17513750802376313. Google Scholar

[74]

E. Venturino, Epidemics in predator-prey models: Disease in the prey,, In Arino, 1 (1995), 381. doi: 10.1093/imammb19.3.185. Google Scholar

[75]

E. Venturino, Epidemics in predator-prey models: Disease in the predators,, IMA Journal of Mathematics Applied in Medicine and Biology, 19 (2002), 185. doi: 10.1093/imammb19.3.185. Google Scholar

[76]

G. A. K. v. Voorn, L. Hemerik, M. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitaion in predator-prey systems with a strong Allee effect,, Mathematical Biosciences, 209 (2007), 451. doi: 10.1016/j.mbs.2007.02.006. Google Scholar

[77]

J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey,, Journal of Mathematical Biology, 62 (2011), 291. doi: 10.1007/s00285-010-0332-1. Google Scholar

[78]

M. Wang, M. Kot and M. Neubert, Integrodifference equations, Allee effects, and invasions,, Journal of Mathematical Biology, 44 (2002), 150. doi: 10.1007/s002850100116. Google Scholar

[79]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,, Texts in Applied Mathematics, 2 (1990). doi: 10.1007/978-1-4757-4067-7. Google Scholar

[80]

Y. Xiao and L. Chen, Modelling and analysis of a predator-prey model with disease in the prey,, Mathematical Biosciences, 171 (2001), 59. doi: 10.1016/S0025-5564(01)00049-9. Google Scholar

[81]

A. A. Yakubu, Allee effects in a discrete-time SIS epidemic model with infected newborns,, Journal of Difference Equations and Applications, 13 (2007), 341. doi: 10.1080/10236190601079076. Google Scholar

[82]

S. R. Zhou, C. Z. Liu and G. Wang, The competitive dynamics of metapopulation subject to the Allee-like effect,, Theoretical Population Biology, 65 (2004), 29. doi: 10.1016/j.tpb.2003.08.002. Google Scholar

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