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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 |
References:
[1] |
W. C. Allee, Animal Aggregations. A Study in General Sociology, University of Chicago Press, Chicago, 1931.
doi: 10.5962/bhl.title.7313. |
[2] |
L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Mathematical Biosciences, 152 (1998), 63-85.
doi: 10.1016/S0025-5564(98)10018-4. |
[3] |
P. Amarasekare, Interactions between local dynamics and dispersal: Insights from single species models, Theoretical Population Biology, 53 (1998), 44-59.
doi: 10.1006/tpbi.1997.1340. |
[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), 1082-1091.
doi: 10.1111/j.1523-1739.2007.00721.x. |
[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), 10-25.
doi: 10.1016/j.jtbi.2007.05.005. |
[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), 147-153.
doi: 10.1017/S0950268802007148. |
[7] |
E. Beltrami and T. O. Carroll, Modelling the role of viral disease in recurrent phytoplankton blooms, Journal of Mathematical Biology, 32 (1994), 857-863.
doi: 10.1007/BF00168802. |
[8] |
E. Beretta and Y. Kuang, Modelling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76.
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), 1821-1839.
doi: 10.1137/080730603. |
[10] |
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Massachusetts, Boston, 1982. |
[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), 375-394.
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), 235-245.
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), 747-766.
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), 15-28.
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), 295-302.
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), 199-211.
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), 230-243.
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, Oxford, 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), 405-410.
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), 277-285.
doi: 10.1017/S1367943000001001. |
[21] |
F. Courchamp, B. Grenfell and T. Clutton-Brock, Impact of natural enemies on obligately cooperatively breeders, Oikos, 91 (2000), 311-322.
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), 941-958.
doi: 10.1080/17513758.2012.697196. |
[23] |
A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos, 112 (2006), 667-679.
doi: 10.1111/j.0030-1299.2006.14243.x. |
[24] |
J. Drake, Allee effects and the risk of biological invasion, Risk Analysis, 24 (2004), 795-802.
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), 549-560.
doi: 10.2307/3546819. |
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H. I. Freedman, A model of predator-prey dynamics as modified by the action of parasite, Mathematical Biosciences, 99 (1990), 143-155.
doi: 10.1016/0025-5564(90)90001-F. |
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A. Friedman and A. A. Yakubu, Fatal disease and demographic allee effect: Population persistence and extinction, Journal of Biological Dynamics, 6 (2012), 495-508.
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), 801-810.
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), 487-496.
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), 539-547.
doi: 10.1006/jtbi.1996.0358. |
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J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983. |
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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, Cambridge, 1995. |
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K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, Journal of Mathematical Biology, 27 (1989), 609-631.
doi: 10.1007/BF00276947. |
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H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 499-653.
doi: 10.1137/S0036144500371907. |
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H. W. Hethcote, W. Wang, L. Han and Z. Ma, A predator-prey model with infected prey, Theoretical Population Biology, 66 (2004), 259-268.
doi: 10.1016/j.tpb.2004.06.010. |
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show all references
References:
[1] |
W. C. Allee, Animal Aggregations. A Study in General Sociology, University of Chicago Press, Chicago, 1931.
doi: 10.5962/bhl.title.7313. |
[2] |
L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Mathematical Biosciences, 152 (1998), 63-85.
doi: 10.1016/S0025-5564(98)10018-4. |
[3] |
P. Amarasekare, Interactions between local dynamics and dispersal: Insights from single species models, Theoretical Population Biology, 53 (1998), 44-59.
doi: 10.1006/tpbi.1997.1340. |
[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), 1082-1091.
doi: 10.1111/j.1523-1739.2007.00721.x. |
[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), 10-25.
doi: 10.1016/j.jtbi.2007.05.005. |
[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), 147-153.
doi: 10.1017/S0950268802007148. |
[7] |
E. Beltrami and T. O. Carroll, Modelling the role of viral disease in recurrent phytoplankton blooms, Journal of Mathematical Biology, 32 (1994), 857-863.
doi: 10.1007/BF00168802. |
[8] |
E. Beretta and Y. Kuang, Modelling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76.
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), 1821-1839.
doi: 10.1137/080730603. |
[10] |
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Massachusetts, Boston, 1982. |
[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), 375-394.
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), 235-245.
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), 747-766.
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), 15-28.
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), 295-302.
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), 199-211.
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), 230-243.
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, Oxford, 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), 405-410.
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), 277-285.
doi: 10.1017/S1367943000001001. |
[21] |
F. Courchamp, B. Grenfell and T. Clutton-Brock, Impact of natural enemies on obligately cooperatively breeders, Oikos, 91 (2000), 311-322.
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), 941-958.
doi: 10.1080/17513758.2012.697196. |
[23] |
A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos, 112 (2006), 667-679.
doi: 10.1111/j.0030-1299.2006.14243.x. |
[24] |
J. Drake, Allee effects and the risk of biological invasion, Risk Analysis, 24 (2004), 795-802.
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), 549-560.
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), 143-155.
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), 495-508.
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), 801-810.
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), 487-496.
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), 539-547.
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, Cambridge, 1995. |
[33] |
K. P. Hadeler and H. I. Freedman, Predator-prey populations with parasitic infection, Journal of Mathematical Biology, 27 (1989), 609-631.
doi: 10.1007/BF00276947. |
[34] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 499-653.
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), 259-268.
doi: 10.1016/j.tpb.2004.06.010. |
[36] |
F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, Journal of Biological Dynamics, 4 (2010), 86-101.
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