Population collapse to extinction: The catastrophic combination of parasitism and Allee effect

Journal of Biological Dynamics

Volumn 4, Issue 1, 2010, Pages 86-101

  Hilker, Frank M ^a,b  

^a UNIVERSITY OF LISBON   (Portugal)

^b University of Bath   (United Kingdom)

Author keywords

Allee effect; Epizootiological model; Fold catastrophe; Species disappearance; Wildlife disease

Indexed keywords

ANIMAL; ARTICLE; BASIC REPRODUCTION NUMBER; BIOLOGICAL MODEL; PARASITE; PHYSIOLOGY; POPULATION DENSITY; POPULATION DYNAMICS; SPECIES EXTINCTION; STATISTICAL MODEL; TIME;

ANIMALS; BASIC REPRODUCTION NUMBER; EXTINCTION, BIOLOGICAL; LOGISTIC MODELS; MODELS, BIOLOGICAL; PARASITES; POPULATION DENSITY; POPULATION DYNAMICS; TIME FACTORS;

EID: 78651247833     PISSN: 17513758     EISSN: 17513766     Source Type: Journal    
DOI: 10.1080/17513750903026429     Document Type: Article

References (61)

1. S. Altizer, C.L. Nunn, P.H. Thrall, J.L. Gittleman, J. Antonovics, A.A. Cunningham, A.P. Dobson, V. Ezenwa, K.E. Jones, A.B. Pedersen, M. Poss, and J.R.C. Pulliam, Social organization and parasite risk in mammals: Integrating theory and empirical studies, Annu. Rev. Ecol. Evol. Syst. 34 (2003), pp. 517-547.
2. R.M. Anderson, Parasite pathogenicity and the depression of host population equilibria, Nature 279 (1979), pp. 150-152.
3. R.M. Anderson and R.M. May, Population biology of infectious diseases: Part I, Nature 280 (1979), pp. 361-367.
4. R.M. Anderson, H.C. Jackson, R.M. May, and A.M. Smith, Population dynamics of fox rabies in Europe, Nature 289 (1981), pp. 765-770.
5. R.M. Anderson, R.M. May, and A.R. McLean, Possible demographic consequences of AIDS in developing countries, Nature 332 (1988), pp. 228-234.
6. J. Arino, C.C. McCluskey, and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math. 64 (2003), pp. 260-276.
7. 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, Epidemiol. Infect. 129 (2002), pp. 147-153.
8. F. Brauer, Models for the spread of universally fatal diseases, J. Math. Biol. 28 (1990), pp. 451-462.
9. B.W. Brook, N.S. Sodhi, and C.J.A. Bradshaw, Synergies among extinction drivers under global change, Trends Ecol. Evol. 23 (2008), pp. 453-460.
10. S.N. Busenberg and K.P. Hadeler, Demography and epidemics, Math. Biosci. 101 (1990), pp. 63-74.
11. S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol. 28 (1990), pp. 257-270.
12. F. Courchamp, L. Berec, and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, NewYork, 2008.
13. P. Daszak, L. Berger, A.A. Cunningham, A.D. Hyatt, D.E. Green, and R. Speare, Emerging infectious diseases and amphibian population declines, Emerg. Infect. Dis. 5 (1999), pp. 735-748.
14. F. de Castro and B. Bolker, Mechanisms of disease-induced extinction, Ecol. Lett. 8 (2005), pp. 117-126.
15. A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos 112 (2006), pp. 667-679.
16. A. Dhooge, W. Govaerts, and Y.A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Softw. 29 (2003), pp. 141-164.
17. O. Diekmann and M. Kretzschmar, Patterns in the effects of infectious diseases on population growth, J.Math. Biol. 29 (1991), pp. 539-570.
18. A. Dobson and P.J. Hudson, Parasites, disease and the structure of ecological communities, Trends Ecol. Evol. 1 (1986), pp. 11-15.
19. A. Dobson and J. Poole, Conspecific aggregation and conservation biology, in Behavioral Ecology and Conservation Biology, T.M. Caro, ed., Oxford University Press, Oxford, 1998, pp. 193-208.
20. E. Doedel, AUTO: A program for the automatic analysis of autonomous systems, Congr. Numer. 30 (1981), pp. 265-284.
21. J. Dushoff, W. Huang, and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol. 36 (1998), pp. 227-248.
22. B. Ermentrout, Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students, SIAM, Philadelphia, 2002.
23. E. Fromont, D. Pontier, and M. Langlais, Dynamics of a feline retrovirus (FeLV) in host populations with variable spatial structure, Proc. R. Soc. Lond. B 265 (1998), pp. 1097-1104.
24. L.Q. Gao and H.W. Hethcote, Disease transmission models with density-dependent demographics, J. Math. Biol. 30 (1992), pp. 717-731.
25. W.M. Getz and J. Pickering, Epidemic models - thresholds and population regulation, Am. Nat. 121 (1983), pp. 892-898.
26. M.G.M. Gomes, P. Rodrigues, F.M. Hilker, N.B. Mantilla-Beniers, M. Muehlen, A.C. Paulo, and G.F. Medley, Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions, J. Theor. Biol. 248 (2007), pp. 608-617.
27. D. Greenhalgh, Hopf bifurcation in epidemic models with a latent period and non-permanent immunity, Math. Comput. Modelling 25 (1997), pp. 85-107.
28. B.T. Grenfell and A.P. Dobson (eds.), Ecology of Infectious Diseases in Natural Populations, Cambridge University Press, Cambridge, 1995.
29. K.P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Math. Biosci. 128 (1995), pp. 41-55.
30. C.D. Harvell, C.E. Mitchell, J.R. Ward, S. Altizer, A.P. Dobson, R.S. Ostfeld, and M.D. Samuel, Climate warming and disease risks for terrestrial and marine biota, Science 296 (2002), pp. 2158-2162.
31. H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), pp. 599-653.
32. [F.M. Hilker, Epidemiological models with demographic Allee effect, in BIOMAT 2008 - International Symposium on Mathematical and Computational Biology, R.P. Mondiani ed., World Scientific Co. Pvt. Ltd., 2009, submitted for publication.
33. F.M. Hilker and K. Schmitz, Disease-induced stabilization of predator-prey oscillations, J. Theor. Biol. 255 (2008), pp. 299-306.
34. F.M. Hilker, M.A. Lewis, H. Seno, M. Langlais, and H. Malchow, Pathogens can slow down or reverse invasion fronts of their hosts, Biol. Invasions 7 (2005), pp. 817-832.
35. F.M. Hilker, M. Langlais, S.V. Petrovskii, and H. Malchow, A diffusive SI model with Allee effect and application to FIV, Math. Biosci. 206 (2007), pp. 61-80.
36. F.M. Hilker, M. Langlais, and H. Malchow, The Allee effect and infectious diseases: Extinction, multistability, and the (dis-)appearance of oscillations, Am. Nat. 173 (2009), pp. 72-88.
37. P.J. Hudson, A. Rizzoli, B.T. Grenfell, H. Heesterbeek, and A.P. Dobson, The Ecology ofWildlife Diseases, Oxford University Press, Oxford, 2001.
38. P.J. Hudson, A.P. Dobson, and K.D. Lafferty, Is a healthy ecosystem one that is rich in parasites?, Trends Ecol. Evol. 21 (2006), pp. 381-385.
39. J.A. Jacquez, C.P. Simon, J. Koopman, L. Sattenspiel, and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patterns, Math. Biosci. 92 (1989), pp. 119-199.
40. C.M. Kribs-Zaleta and J.X. Velasco-Hernández, A simple vaccination model with multiple endemic states, Math. Biosci. 164 (2000), pp. 183-201.
41. K.D. Lafferty, A. Allesina, M. Arim, C.J. Briggs, G. De Leo, A.P. Dobson, J.A. Dunne, P.T.J. Johnson, A.M. Kuris, D.J. Marcogliese, N.D. Martinez, J. Memmott, P.A. Marquet, J.P. McLaughlin, E.A. Mordecai, M. Pascual, R. Poulin, and D.W. Thieltges, Parasites in food webs: The ultimate missing links, Ecol. Lett. 11 (2008), pp. 533-546.
42. M. Martcheva and H.R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with superinfection, J. Math. Biol. 46 (2003), pp. 385-424.
43. H. McCallum, N. Barlow, and J. Hone, How should pathogen transmission be modelled?, Trends Ecol. Evol. 16 (2001), pp. 295-300.
44. G.F. Medley, N.A. Lindop, W.J. Edmunds, and D.J. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control, Nat. Med. 7 (2001), pp. 619-624.
45. E.J. Milner-Gulland, O.M. Bukreeva, T. Coulson, A.A. Lushchekina, M.V. Kholodova, A.B. Bekenov, and I.A. Grachev, Conservation: Reproductive collapse in saiga antelope harems, Nature 422 (2003), p. 135.
46. S.M. Moghadas, Analysis of an epidemic model with bistable equilibria using the Poincaré index, Appl. Math. Comput. 149 (2004), pp. 689-702.
47. P.K. Molnár, A.E. Derocher, M.A. Lewis, and M.K. Taylor, Modelling the mating system of polar bears:Amechanistic approach to the Allee effect, Proc. R. Soc. Lond. B 275 (2008), pp. 217-226.
48. A. Pugliese, Population models for diseases with no recovery, J. Math. Bio. 28 (1990), pp. 65-82.
49. M.G. Roberts, The pluses and minuses of R0, J. R. Soc. Interface 4 (2007), pp. 949-961.
50. B.M. Rothschild and R. Laub, Hyperdisease in the late Pleistocene: Validation of an early 20th century hypothesis, Naturwissenschaften 93 (2006), pp. 557-564.
51. S. Rowe, J.A. Hutchings, D. Bekkevold, and A. Rakitin, Depensation, probability of fertilization, and the mating system of Atlantic cod (Gadus morhua L.), ICES J. Mar. Sci. 61 (2004), pp. 1144-1150.
52. M. Safan, H. Heesterbeek, and K. Dietz, The minimum effort required to eradicate infections in models with backward bifurcation, J. Math. Bio. 53 (2006), pp. 703-718.
53. B.H. Singer and D.E. Kirschner, Influence of backward bifurcation on interpretation of R0 in a model of epidemic tuberculosis with reinfection, Math. Biosci. Eng. 1 (2004), pp. 81-93.
54. K.F. Smith, D.F. Sax, and K.D. Lafferty, Evidence for the role of infectious disease in species extinction and endangerment, Conserv. Biol. 20 (2006), pp. 1349-1357.
55. P.A. Stephens, W.J. Sutherland, and R.P. Freckleton, What is the Allee effect?, Oikos 87 (1999), pp. 185-190.
56. H.R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci. 111 (1992), pp. 99-130.
57. H.R. Thieme, T. Dhirasakdanon, Z. Han, and R. Trevino, Species decline and extinction: Synergy of infectious diseases and Allee effect?, J. Bio. Dyn. 3 (2009), pp. 305-323.
58. P.H. Thrall, J. Antonovics, and D.W. Hall, Host and pathogen coexistence in sexually transmitted and vector-borne diseases characterized by frequency-dependent disease transmission, Am. Nat. 142 (1993), pp. 543-552.
59. P. van den Driessche and J. Watmough, A simple SIS model with a backward bifurcation, J. Math. Biol. 40 (2000), pp. 525-540.
60. [K.B. Wyatt, P.F. Campos, M.T.P. Gilbert, S.O. Kolokotronis, W.H. Hynes, R. DeSalle, S.J. Ball, P. Daszak, R.D.E. MacPhee, and A.D. Greenwood, Historical mammal extinction on Christmas Island (Indian Ocean) correlates with introduced infectious disease, PLoS One 3 (2008), p. e3602.
61. J. Zhou and H.W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Biol. 32 (1994), pp. 809-834.