Population extinction in deterministicand stochastic discrete-time epidemic models with periodic coefficients with applications to amphibian populations

Natural Resource Modeling

Volumn 19, Issue 2, 2006, Pages 117-164

  Emmert, Keith E ^a   Allen, Linda J S ^b  

^a TARLETON STATE UNIVERSITY   (United States)

^b TEXAS TECH UNIVERSITY   (United States)

Author keywords

Amphibians; Branching process; Si epidemic; Structured model

Indexed keywords

EID: 53349149654     PISSN: 08908575     EISSN: 19397445     Source Type: Journal    
DOI: 10.1111/j.1939-7445.2006.tb00178.x     Document Type: Article

References (41)

1. L.J.S. Allen [2005], An Introduction to Deterministic Models in Biology, Prentice Hall, Upper Saddle River, NJ.
2. L.J.S. Allen, D.A. Flores, R.K Ratnayake and J.R. Herbold [2002], Discrete-Time Deterministic and Stochastic Models for the Spread of Rabies, Appl. Math. Comp. 132, 271–292.
3. L. Berger, R. Speare, P. Daszak, D.E. Green, A.A. Cunningham, C.L. Goggin, R. Slocombe, M.A. Ragan, Alex D. Hyatt, K.R. McDonald, H.B. Hines, K.R. Lips, G. Marantelli and H. Parkes [1998], Chytridiomycosis Causes Amphibian Mortality Associated with Population Declines in the Rain Forests of Australia and Central America, Proc. Natl. Acad. Sci. USA, 95, 9031–9036.
4. L. Berger, R. Speare and A. Hyatt [1999], Chytrid Fungi and Amphibian Declines: Overview, Implications and Future Directions, in Declines and Disappearances of Australian Frogs (A. Campbell, ed.), Environment Australia, Canberra, pp. 21–31.
5. L. Berger, R. Speare, H.B. Hines, G. Marantelli, A.D. Hyatt, K.R. McDonald, L.F. Skerratt, V. Olsen, J.M. Clarke, G. Gillespie, M. Mahony, N. Sheppard, C. Williams and M.J. Tyler [2004], Effect of Season and Temperature on Mortality in Amphibians Due to Chytridiomycosis, Austral. Vet. J. 82, 31–36.
6. J. Bosch, I. Martínez-Solano and M. García-París [2001], Evidence of Chytrid Fungus Infection Involved in the Decline of the Common Midwife Toad (Alytex obstetricans) in Protected Areas of Central Spain, Biol. Conserv. 97, 331–337.
7. C. Carey, N. Cohen and L. Rollins-Smith [1999], Amphibian Declines: An Im-munological Perspective, Dev. Comp. Immunol. 23, 459–472.
8. C. Carey, A.P. Pessier and A.D. Peace [2003], Pathogens, Infectious Disease, and Immune Defenses, in Amphibian Conservation, Smithsonian Books, Washington, London, 2003, pp. 127–136.
9. H. Caswell [2001], Matrix Population Models, Sinauer Assoc., Inc. Pub., Sunder-land, MA.
10. J.P. Collins and A. Storfer [2003], Global Amphibian Declines: Sorting the Hypotheses, Diversity and Distributions 9, 89–98.
11. P. Cull [1986], Local and Global Stability for Population Models, Biol. Cyber. 54, 141–149.
12. J.M. Cushing [1998], An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia.
13. P. Daszak, L. Berger, A.A. Cunningham, A.D. Hyatt, D.E. Green and R. Speare [1999], Emerging Infectious Diseases and Amphibian Population Declines, Emerging Infect. Diseases 5, 735–748.
14. P. Daszak, A.A. Cunningham and A.D. Hyatt [2000], Emerging Infectious Diseases of Wildlife Threats to Biodiversity and Human Health, Science 287, 443–449.
15. E.W. Davidson, M. Parris, J.P. Collins, J.E. Longcore, A.P. Pessier and J. Brunner [2003], Pathogenicity and Transmission of Chytridiomycosis in Tiger Salamanders (Ambystoma triginum), Copeia 3, 601–607.
16. C.K. Dodd, Jr. and R.B. Bury [2002], Amphibians at a Crossroads: Overview, People, Land and Water, August, p. 28.
17. A.J. Drew [2004], An Investigation of Climatic and Geographic Factors on the Growth and Spread of the Chytrid Fungus on Amphibian Populations in Australia, M.S. Thesis, Texas Tech University, Lubbock, TX.
18. L. Edelstein-Keshet [1988], Mathematical Models in Biology, McGraw Hill, New York.
19. S.N. Elaydi [1999], An Introduction to Difference Equations, 2nd ed., Springer-Verlag, New York.
20. K.E. Emmert [2004], Deterministic and Stochastic Discrete-Time Epidemic Models with Applications to Amphibians, Ph.D. Dissertation, Texas Tech University.
21. K.E. Emmert and L.J.S. Allen [2004], Population Persistence and Extinction in a Discrete-Time, Stage-Structured Epidemic Model, J. Difference Equ. Appl. 10, 1177–1199.
22. T.E. Harris [1963], The Theory of Branching Processes, Prentice Hall, Inc., Englewood Cliffs, NJ.
23. G. Högnäs [1997], On the Quasi-Stationary Distribution of a Stochastic Ricker Model, Stoch. Proc. Appl. 70, 243–263.
24. G. Högnäs [2000], On Some One-Dimensional Stochastic Population Models, Contemp. Math. vol. 261, Amer. Math. Soc., Providence, RI, pp. 209–220.
25. J.E. Houlahan, G.S. Findlay, B.R. Schmidt, A.H. Meyers and S.L. Kuzmin [2000], Quantitative Evidence for Global Amphibian Population Declines, Nature 404, 752–755.
26. P. Jagers [1975], Branching Processes with Biological Applications, Wiley, Chich-ester.
27. M. Johnson and R. Speare [2003], Survival of Batrachocytrium dendrobatidis in Water: Quarantine and Disease Control Implications, Emerging Infect. Diseases 9, 922–925.
28. J. Kiesecker, L. Belden, K. Shea and M. Rubbo [2004], Amphibian Decline and Emerging Disease, Amer. Scientist 92, 138–147.
29. R. Kon, Y. Saito and Y. Takeuchi [2003], Permanence of Single-Species Stage-Structured Models, J. Math. Biol. 48, 515–528.
30. K.R. Lips [1999], Mass Mortality and Population Declines of Anurans at an Upland Site in Western Panama, Conserv. Biol. 13, 117–125.
31. J.E. Longcore, A.P. Pessier and D.K. Nichols [1999], Batrachochytrium dendroba-tidis gen. et sp. nov., A Chytrid Pathogenic to Amphibians, Mycologia 91, 219–227.
32. I. Martínez-Solano, J. Bosch and M. García-París [2003], Demographic Trends and Community Stability in a Montane Amphibian Assemblage, Conserv. Biol. 17, 238–244.
33. C.C. McCluskey and J.S. Muldowney [1998], Bendixson-Dulac Criteria for Difference Equations, J. Dynam. Differential Equations 10, 567–576.
34. C.J. Mode [1971], Multitype Branching Processes Theory and Applications, Elsevier, New York.
35. V. Morell [1999], Are Pathogens Felling Frogs ?, Science 284, 728–731.
36. E.A. Morehouse, T.Y. James, A.R.D. Ganley, R. Vilgalys, L. Berger, P.J. Murphy and J.E. Longcore [2003], Multilocus Sequence Typing Suggests the Chytrid Pathogen of Amphibians is a Recently Emerged Clone, Molecular Ecol. 12, 395–403.
37. S.S. Morse [1995], Factors in the Emergence of Infectious Diseases, Emerging Infect. Diseases 1, 7–15.
38. R.A. Relyea and N. Mills [2001], Predator-Induced Stress Makes the Pesticide Carbaryl More Deadly to Gray Treefrog Tadpoles (Hyla versicolor), Proc. Natl. Acad. Sci. USA 98, 2491–2496.
39. J.L. Sabo, E.E. Holmes and P. Kareiva [2004], Efficacy Simple Viability Models in Ecological Risk Assessment: Does Density Dependence Matter ?, Ecology 85, 328–341.
40. R.D. Semlitsch [2003], Amphibian Conservation, Smithsonian Books, Washington, London.
41. S. Tang and L. Chen [2003], The Effect of Seasonal Harvesting on Stage-Structured Population Models, J. Math. Biol. 48, 357–374.