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Bifurcations of an SIRS epidemic model with nonlinear incidence rate
Allee effects in an iteroparous host population and in host-parasitoid interactions
1. | Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States |
References:
[1] |
W. C. Allee, "The Social Life of Animals," William Heinemann, London, 1938. |
[2] |
L. J. S. Allen, "An Introduction to Mathematical Biology," Prentice Hall, New Jersey, 2006. |
[3] |
M. Begon, J. Harper and C. Townsend, "Ecology: Individuals, Populations and Communities," Blackwell Science Ltd, New York, 1996. |
[4] |
W. J. Bell and K. G. Adiyodi, "The American Cockroach," Chapman and Hall, London, 1981. |
[5] |
A. Brännström and D. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B, 272 (2005), 2065-2072.
doi: doi:10.1098/rspb.2005.3185. |
[6] |
J. B. Conway, "Functions of One Complex Variable," 2nd edition, Springer, New York, 1978. |
[7] |
F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford University Press, New York, 2008.
doi: doi:10.1093/acprof:oso/9780198570301.001.0001. |
[8] |
J. M. Cushing, The Allee effect in age-structured population dynamics, in "Mathematical Ecology" (eds. T. Hallam, L. Gross and S. Levin), (1988), 479-505. |
[9] |
J. M. Cushing, A strong ergodic theorem for some nonlinear matrix models for the dynamics of structured population, Nat. Res. Mod., 3 (1989), 331-357. |
[10] |
J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333. |
[11] |
J. M. Cushing, "An Introduction to Structured Population Models," Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, 1998. |
[12] |
J. M. Cushing, Nonlinear semelparous Leslie models, Math. Bio. Sci. Eng., 3 (2006), 17-36. |
[13] |
B. Dennis, Allee effects, population growth, critical density, and the chance of extinction, Nat. Res. Mod., 3 (1989), 481-538. |
[14] |
B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401.
doi: doi:10.1034/j.1600-0706.2002.960301.x. |
[15] |
A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos, 112 (2006), 667-679.
doi: doi:10.1111/j.0030-1299.2006.14243.x. |
[16] |
A. Grove and G. Ladas, "Periodicities in Nonlinear Difference Equations," CRC Press, Boca Raton, 2005. |
[17] |
J. Hale and H. Kocak, "Dynamics and Bifurcations," Springer-Verlag, New York, 1991. |
[18] |
F. M. Hilker, M. Langlais, S. V. Petrovskii and H. Malcho, A diffusive SI model with Allee effect and application to FIV, Math. Biosci., 206 (2007), 61-80.
doi: doi:10.1016/j.mbs.2005.10.003. |
[19] |
F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Bio. Dyn., 4 (2010), 86-101.
doi: doi:10.1080/17513750903026429. |
[20] |
M. Hirsch and H. Smith, Monotone maps: A review, J. Diff. Equ. & Appl., 11 (2005), 379-398. |
[21] |
J. Hofbauer and J. So, Uniform persistence and repellors for maps, Proc. Am. Math. Soc., 107 (1989), 1137-1142. |
[22] |
S. R.-J. Jang, Allee effects in a discrete-time host parasitoid model with stage structure in the host, Dis. Con. Dyn. Sys. Ser. B, 8 (2007), 145-159.
doi: doi:10.3934/dcdsb.2007.8.145. |
[23] |
S. R.-J. Jang, Discrete host-parasitoid models with Allee effects and age structure in the host, Math. Biosci. & Eng., 7 (2010), 67-81.
doi: doi:10.3934/mbe.2010.7.67. |
[24] |
S. R.-J. Jang, Dynamics of an age-structured population with Allee effects and harvesting, J. Bio. Dyn., 4 (2010), 409-427.
doi: doi:10.1080/17513750903389082. |
[25] |
M. Kulenovic and M. Nurkanovic, Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation, J. Diff. Equ. & Appl., 9 (2003), 149-159. |
[26] |
M. Kulenovic and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Int. J. Bif. Chaos, to appear. |
[27] |
R. M. May, H. P. Hassell, R. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, J. Ani. Ecol., 50 (1981), 855-865.
doi: doi:10.2307/4142. |
[28] |
C. C. McCluskey and J. S. Muldowney, Bendixson-Dulac criteria for difference equations, J. Dyn. Dif. Eq., 10 (1998), 567-575.
doi: doi:10.1023/A:1022677008393. |
[29] |
A. Morozov, S. Petrovskii and B.-L. Li, Bifurcations and chaos in a predator-prey system with the Allee effect, Proc. Roy. Soc., Ser. B, 271 (2004), 1407-1414.
doi: doi:10.1098/rspb.2004.2733. |
[30] |
A. J. Nicholson and V. A. Bailey, The balance of animal population: Part I, Proc. Zool. Soc. Lond., 3 (1935), 551-598. |
[31] |
W. F. Patterson, J. H. Cowan, G. R. Fitzhugh and D. L. Nieland, "Population Ecology and Fisheries of U. S. Gulf of Mexico Red Snapper," American Fisheries Society, Bethesda, Maryland, 2007. |
[32] |
C. Robinson, "Stability, Symbolic Dynamics, and Chaos," CRC Press, Boca Raton, 1995. |
[33] |
S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theo. Pop. Bio., 64 (2003), 201-209.
doi: doi:10.1016/S0040-5809(03)00072-8. |
[34] |
H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. & Appl., 3 (1998), 335-357. |
[35] |
D. R. Suiter, Biological suppression of synanthropic cockroaches, J. Agri. Entomo., 14 (1997), 259-270. |
[36] |
H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious diseases and Allee effect, J. Biol. Dyn., 3 (2009), 305-323.
doi: doi:10.1080/17513750802376313. |
[37] |
G. Voorn, L. Hemerik, M. Boer and B. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469.
doi: doi:10.1016/j.mbs.2007.02.006. |
[38] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer, New York, 2003. |
[39] |
S. Zhou and G. Wang, Allee-like effects in metapopulation dynamics, Math. Biosci., 189 (2004), 103-113.
doi: doi:10.1016/j.mbs.2003.06.001. |
[40] |
S. Zhou, Y. Liu and G. Yang, The stability of predator-prey systems subject to the Allee effects, Theo. Pop. Biol., 67 (2005), 23-31.
doi: doi:10.1016/j.tpb.2004.06.007. |
show all references
References:
[1] |
W. C. Allee, "The Social Life of Animals," William Heinemann, London, 1938. |
[2] |
L. J. S. Allen, "An Introduction to Mathematical Biology," Prentice Hall, New Jersey, 2006. |
[3] |
M. Begon, J. Harper and C. Townsend, "Ecology: Individuals, Populations and Communities," Blackwell Science Ltd, New York, 1996. |
[4] |
W. J. Bell and K. G. Adiyodi, "The American Cockroach," Chapman and Hall, London, 1981. |
[5] |
A. Brännström and D. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B, 272 (2005), 2065-2072.
doi: doi:10.1098/rspb.2005.3185. |
[6] |
J. B. Conway, "Functions of One Complex Variable," 2nd edition, Springer, New York, 1978. |
[7] |
F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford University Press, New York, 2008.
doi: doi:10.1093/acprof:oso/9780198570301.001.0001. |
[8] |
J. M. Cushing, The Allee effect in age-structured population dynamics, in "Mathematical Ecology" (eds. T. Hallam, L. Gross and S. Levin), (1988), 479-505. |
[9] |
J. M. Cushing, A strong ergodic theorem for some nonlinear matrix models for the dynamics of structured population, Nat. Res. Mod., 3 (1989), 331-357. |
[10] |
J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333. |
[11] |
J. M. Cushing, "An Introduction to Structured Population Models," Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, 1998. |
[12] |
J. M. Cushing, Nonlinear semelparous Leslie models, Math. Bio. Sci. Eng., 3 (2006), 17-36. |
[13] |
B. Dennis, Allee effects, population growth, critical density, and the chance of extinction, Nat. Res. Mod., 3 (1989), 481-538. |
[14] |
B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401.
doi: doi:10.1034/j.1600-0706.2002.960301.x. |
[15] |
A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos, 112 (2006), 667-679.
doi: doi:10.1111/j.0030-1299.2006.14243.x. |
[16] |
A. Grove and G. Ladas, "Periodicities in Nonlinear Difference Equations," CRC Press, Boca Raton, 2005. |
[17] |
J. Hale and H. Kocak, "Dynamics and Bifurcations," Springer-Verlag, New York, 1991. |
[18] |
F. M. Hilker, M. Langlais, S. V. Petrovskii and H. Malcho, A diffusive SI model with Allee effect and application to FIV, Math. Biosci., 206 (2007), 61-80.
doi: doi:10.1016/j.mbs.2005.10.003. |
[19] |
F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Bio. Dyn., 4 (2010), 86-101.
doi: doi:10.1080/17513750903026429. |
[20] |
M. Hirsch and H. Smith, Monotone maps: A review, J. Diff. Equ. & Appl., 11 (2005), 379-398. |
[21] |
J. Hofbauer and J. So, Uniform persistence and repellors for maps, Proc. Am. Math. Soc., 107 (1989), 1137-1142. |
[22] |
S. R.-J. Jang, Allee effects in a discrete-time host parasitoid model with stage structure in the host, Dis. Con. Dyn. Sys. Ser. B, 8 (2007), 145-159.
doi: doi:10.3934/dcdsb.2007.8.145. |
[23] |
S. R.-J. Jang, Discrete host-parasitoid models with Allee effects and age structure in the host, Math. Biosci. & Eng., 7 (2010), 67-81.
doi: doi:10.3934/mbe.2010.7.67. |
[24] |
S. R.-J. Jang, Dynamics of an age-structured population with Allee effects and harvesting, J. Bio. Dyn., 4 (2010), 409-427.
doi: doi:10.1080/17513750903389082. |
[25] |
M. Kulenovic and M. Nurkanovic, Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation, J. Diff. Equ. & Appl., 9 (2003), 149-159. |
[26] |
M. Kulenovic and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Int. J. Bif. Chaos, to appear. |
[27] |
R. M. May, H. P. Hassell, R. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, J. Ani. Ecol., 50 (1981), 855-865.
doi: doi:10.2307/4142. |
[28] |
C. C. McCluskey and J. S. Muldowney, Bendixson-Dulac criteria for difference equations, J. Dyn. Dif. Eq., 10 (1998), 567-575.
doi: doi:10.1023/A:1022677008393. |
[29] |
A. Morozov, S. Petrovskii and B.-L. Li, Bifurcations and chaos in a predator-prey system with the Allee effect, Proc. Roy. Soc., Ser. B, 271 (2004), 1407-1414.
doi: doi:10.1098/rspb.2004.2733. |
[30] |
A. J. Nicholson and V. A. Bailey, The balance of animal population: Part I, Proc. Zool. Soc. Lond., 3 (1935), 551-598. |
[31] |
W. F. Patterson, J. H. Cowan, G. R. Fitzhugh and D. L. Nieland, "Population Ecology and Fisheries of U. S. Gulf of Mexico Red Snapper," American Fisheries Society, Bethesda, Maryland, 2007. |
[32] |
C. Robinson, "Stability, Symbolic Dynamics, and Chaos," CRC Press, Boca Raton, 1995. |
[33] |
S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theo. Pop. Bio., 64 (2003), 201-209.
doi: doi:10.1016/S0040-5809(03)00072-8. |
[34] |
H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. & Appl., 3 (1998), 335-357. |
[35] |
D. R. Suiter, Biological suppression of synanthropic cockroaches, J. Agri. Entomo., 14 (1997), 259-270. |
[36] |
H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious diseases and Allee effect, J. Biol. Dyn., 3 (2009), 305-323.
doi: doi:10.1080/17513750802376313. |
[37] |
G. Voorn, L. Hemerik, M. Boer and B. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469.
doi: doi:10.1016/j.mbs.2007.02.006. |
[38] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer, New York, 2003. |
[39] |
S. Zhou and G. Wang, Allee-like effects in metapopulation dynamics, Math. Biosci., 189 (2004), 103-113.
doi: doi:10.1016/j.mbs.2003.06.001. |
[40] |
S. Zhou, Y. Liu and G. Yang, The stability of predator-prey systems subject to the Allee effects, Theo. Pop. Biol., 67 (2005), 23-31.
doi: doi:10.1016/j.tpb.2004.06.007. |
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