Dynamics of a predatorprey system with prey subject to Allee effects and disease
Pages: 877  918,
Issue 4,
August
2014
doi:10.3934/mbe.2014.11.877 Abstract
References
Full text (2746.9K)
Related Articles
Yun Kang  Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, United States (email)
Sourav Kumar Sasmal  Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India (email)
Amiya Ranjan Bhowmick  Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India (email)
Joydev Chattopadhyay  Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India (email)
1 
W. C. Allee, Animal Aggregations. A Study in General Sociology, University of Chicago Press, Chicago, 1931. 

2 
L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Mathematical Biosciences, 152 (1998), 6385. 

3 
P. Amarasekare, Interactions between local dynamics and dispersal: Insights from single species models, Theoretical Population Biology, 53 (1998), 4459. 

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), 10821091. 

5 
N. Bairagi, P. K. Roy and J. Chattopadhyay, Role of infection on the stability of a predatorprey system with several response functions  A comparative study, Journal of Theoretical Biology, 248 (2007), 1025. 

6 
M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner, A clarification of transmission terms in hostmicroparasite models: Numbers, densities and areas, Epidemiology and Infection, 129 (2002), 147153. 

7 
E. Beltrami and T. O. Carroll, Modelling the role of viral disease in recurrent phytoplankton blooms, Journal of Mathematical Biology, 32 (1994), 857863. 

8 
E. Beretta and Y. Kuang, Modelling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 5776. 

9 
F. S. Berezovskaya, B. Song and C. CastilloChavez, Role of prey dispersal and refuges on predatorprey dynamics, SIAM Journal on Applied Mathematics, 70 (2010), 18211839. 

10 
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Massachusetts, Boston, 1982. 

11 
D. S. Boukal and L. Berec, Singlespecies Models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218 (2002), 375394. 

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), 235245. 

13 
J. Chattopadhyay and O. Arino, A predatorprey model with disease in the prey, Nonlinear Analysis, 36 (1999), 747766. 

14 
J. Chattopadhyay and S. Pal, Viral infection on phytoplanktonzooplankton systema mathematical model, Ecological Modelling, 151 (2002), 1528. 

15 
J. Chattopadhyay, R. Sarkar, M. E. FritzcheHoballah, T. Turlings and L. Bersier, Parasitoids may determine plant fitness  A mathematical model based on experimental data, Journal of Theoretical Biology, 212 (2001), 295302. 

16 
J. Chattopadhyay, P. Srinivasu and N. Bairagi, Pelicans at risk in Salton Seaan ecoepidemiological modelII, Ecological Modelling, 167 (2003), 199211. 

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), 230243. 

18 
F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008. 

19 
F. Courchamp, T. CluttonBrock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology & Evolution, 14 (1999), 405410. 

20 
F. Courchamp, T. CluttonBrock and B. Grenfell, Multipack dynamics and the Allee effect in the African wild dog, Lycaon pictus, Animal Conservation, 3 (2000), 277285. 

21 
F. Courchamp, B. Grenfell and T. CluttonBrock, Impact of natural enemies on obligately cooperatively breeders, Oikos, 91 (2000), 311322. 

22 
J. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects, Journal of Biological Dynamics, 6 (2012), 941958. 

23 
A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos, 112 (2006), 667679. 

24 
J. Drake, Allee effects and the risk of biological invasion, Risk Analysis, 24 (2004), 795802. 

25 
J. Ferdy, F. Austerlitz, J. Moret, P. Gouyon and B. Godelle, Pollinatorinduced density dependence in deceptive species, Oikos, 87 (1999), 549560. 

26 
H. I. Freedman, A model of predatorprey dynamics as modified by the action of parasite, Mathematical Biosciences, 99 (1990), 143155. 

27 
A. Friedman and A. A. Yakubu, Fatal disease and demographic allee effect: Population persistence and extinction, Journal of Biological Dynamics, 6 (2012), 495508. 

28 
J. C. Gascoigne and R. N. Lipccius, Allee effects driven by predation, Journal of Applied Ecology, 41 (2004), 801810. 

29 
M. Groom, Allee effects limit population viability of an annual plant, The American Naturalist, 151 (1998), 487496. 

30 
Y. Gruntfest, R. Arditi and Y. Dombronsky, A fragmented population in a varying environment, Journal of Theoretical Biology, 185 (1997), 539547. 

31 
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, SpringerVerlag, 1983. 

32 
F. M. D. Gulland, The Impact of Infectious Diseases on Wild Animal PopulationsA Review. In: Ecology of Infectious Diseases in Natural Populations, Cambridge University Press, Cambridge, 1995. 

33 
K. P. Hadeler and H. I. Freedman, Predatorprey populations with parasitic infection, Journal of Mathematical Biology, 27 (1989), 609631. 

34 
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 499653. 

35 
H. W. Hethcote, W. Wang, L. Han and Z. Ma, A predatorprey model with infected prey, Theoretical Population Biology, 66 (2004), 259268. 

36 
F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, Journal of Biological Dynamics, 4 (2010), 86101. 

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), 7288. 

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), 6180. 

39 
F. M. Hilker and K. Schmitz, Diseaseinduced stabilization of predatorprey oscillations, Journal of Theoretical Biology, 255 (2008), 299306. 

40 
C. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91 (1959), 385398. 

41 
V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Mathematical Biosciences, 111 (1992), 171. 

42 
J. Jacobs, Cooperation, optimal density and low density thresholds: yet another modification of the logistic model, Oecologia, 64 (1984), 389395. 

43 
S. R. J. Jang, Discretetime hostparasitoid models with Allee effects: Density dependence versus parasitism, Journal of Difference Equations and Applications, 17 (2011), 525539. 

44 
S. R. J. Jang, Allee effects in a discretetime hostparasitoid model, Journal of Difference Equations and Applications, 12 (2006), 165181. 

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. (Under revision). 

46 
Y. Kang, Scramble competitions can rescue endangered species subject to strong Allee effects, Mathematical Biosciences, 241 (2013), 7587. 

47 
Y. Kang, A. R. Bhowmick, S. K. Sasmal and J. Chattopadhyay, Hostparasitoid systems with predationdriven Allee effects in host population, preprint. 

48 
Y. Kang and C. CastilloChavez, Multiscale analysis of compartment models with dispersal, Journal of Biological Dynamics, 6 (2012), 5079. 

49 
Y. Kang and C. CastilloChavez, A Simple Epidemiological Model for Populations in The Wild with Allee Effects and Disease Modified Fitness, Journal of Discrete and Continuous Dynamical SystemsB, 2013, (Accepted). 

50 
Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic twopatch models with Allee effects, Journal of Mathematical Biology, 62 (2011), 925973. 

51 
Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, Journal of Mathematical Biology, 67 (2013), 12271259. 

52 
Y. Kang and A.A. Yakubu, Weak Allee effects and species coexistence, Nonlinear Analysis: Real World Applications, 12 (2011), 33293345. 

53 
M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly, Oikos, 82 (1998), 384392. 

54 
B. Lamont, P. Klinkhamer and E. Witkowski, Population fragmentation may reduce fertility to zero in Banksia goodiidemonstration of the Allee effect, Oecologia, 94 (1993), 446450. 

55 
M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141158. 

56 
H. McCallum, N. Barlow and J. Hone, How should pathogen transmission be modelled? Trends in Ecology & Evolution, 16 (2001), 295300. 

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), 9597. 

58 
V. PadrĂ³n and M. C. Trevisan, Effect of aggregating behavior on population recovery on a set of habitat islands, Mathematical Biosciences, 165 (2000), 6378. 

59 
A. Potapov, E. Merrill and M. A. Lewis, Wildlife disease elimination and density dependence, Proceedings of the Royal Society  Biological Sciences, 279 (2012), 31393145. 

60 
R. Ricklefs and G. Miller, Ecology, Williams and Wilkins Co., Inc., New York, 4th ed., 2000. 

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), 217226. 

62 
S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64 (2003), 201209. 

63 
J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, Journal of Mathematical Biology, 52 (2006), 807829. 

64 
M. Sieber and F. M. Hilker, The hydra effect in predatorprey models, Journal of Mathematical Biology, 64 (2012), 341360. 

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), 153166. 

66 
P. Stephens and W. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401405. 

67 
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect? Oikos, 87 (1999), 185190. 

68 
A. Stoner and M. RayCulp, Evidence for Allee effects in an overharvested marine gastropod: Density dependent mating and egg production, Marine Ecology Progress Series, 202 (2000), 297302. 

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), 23902393. 

70 
M. Su, C. Hui, Y. Zhang and Z. Li, Spatiotemporal dynamics of the epidemic transmission in a predatorprey syatem, Bulletin of Mathematical Biology, 70 (2008), 21952210. 

71 
C. Taylor and A. Hastings, Allee effects in biological invasions, Ecology Letters, 8 (2005), 895908. 

72 
H. R. Thieme, Convergence results and a PoincarĂ©Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755763. 

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), 305323. 

74 
E. Venturino, Epidemics in predatorprey models: Disease in the prey, In Arino, O., Axelrod, D., Kimmel, M., Langlais, M. (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, 1 (1995), 381393. 

75 
E. Venturino, Epidemics in predatorprey models: Disease in the predators, IMA Journal of Mathematics Applied in Medicine and Biology, 19 (2002), 185205. 

76 
G. A. K. v. Voorn, L. Hemerik, M. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitaion in predatorprey systems with a strong Allee effect, Mathematical Biosciences, 209 (2007), 451469. 

77 
J. Wang, J. Shi and J. Wei, Predatorprey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291331. 

78 
M. Wang, M. Kot and M. Neubert, Integrodifference equations, Allee effects, and invasions, Journal of Mathematical Biology, 44 (2002), 150168. 

79 
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, 2, Springer, New York, 1990. 

80 
Y. Xiao and L. Chen, Modelling and analysis of a predatorprey model with disease in the prey, Mathematical Biosciences, 171 (2001), 5982. 

81 
A. A. Yakubu, Allee effects in a discretetime SIS epidemic model with infected newborns, Journal of Difference Equations and Applications, 13 (2007), 341356. 

82 
S. R. Zhou, C. Z. Liu and G. Wang, The competitive dynamics of metapopulation subject to the Alleelike effect, Theoretical Population Biology, 65 (2004), 2937. 

Go to top
