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Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect
1. | Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing, 401331, China |
2. | Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, United States |
References:
[1] |
L. Allen, J. Fagan, G. Hognas and H. Fagerholm, Population extinction in discrete-time stochastic population models with an Allee effect, J. Difference Eq. & Appl., 11 (2005), 273-293.
doi: 10.1080/10236190412331335373. |
[2] |
R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, volume 11 of Fish and Fisheries Series, Chapman and Hall, London, 1957, reprinted, 1993. |
[3] |
J. M. Cushing, The Allee effect in age-structured population dynamics, In T. Hallam, L. Gross, and S. Levin, editors, Mathematical Ecology, pages 479-505. World Sci. Publ., Teaneck, NJ, 1988. |
[4] |
J. M. Cushing, Oscillations in age-structured population models with Allee effect, J. Comput. and Appl. Math., 52 (1994), 71-80.
doi: 10.1016/0377-0427(94)90349-2. |
[5] |
J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations, J. Difference Eq. & Appl., 7 (2001), 857-872.
doi: 10.1080/10236190108808308. |
[6] |
J. M. Cushing and S. M. Henson, A periodically forced Beverton-Holt equation, J. Difference Eq. & Appl., 8 (2002), 1119-1120.
doi: 10.1080/1023619021000053980. |
[7] |
H. G. Davis, C. M. Taylor, J. C. Civille and D. R. Strong, An Allee effect at the front of a plant invasion: Spartina in a Pacific estuary, J. Ecology, 92 (2004), 321-327.
doi: 10.1111/j.0022-0477.2004.00873.x. |
[8] |
B. Dennis, Allee effects: Population growth, critical density and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538. |
[9] |
S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. |
[10] |
S. Elaydi and R. J. Sacker, Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjecture, In S. Elaydi, G. Ladas, B. Aulbach, and O. Dosly, editors, 8th Internat. Conf. on Difference Equations and Appl.(2003), Brno, Czech Republic, pages 113-126. Chapman and Hall, 2005.
doi: 10.1201/9781420034905. |
[11] |
S. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Difference Eq. & Appl., 11 (2005), 337-346.
doi: 10.1080/10236190412331335418. |
[12] |
S. Elaydi and R. J. Sacker, Periodic difference equations, population biology and the Cushing-Henson conjectures, Math. Biosciences, 201 (2006), 195-207.
doi: 10.1016/j.mbs.2005.12.021. |
[13] |
S. Elaydi and R. J. Sacker, Population models with Allee effect, A new model, J. Biological Dynamics, 4 (2010), 397-408.
doi: 10.1080/17513750903377434. |
[14] |
H. Eskola and K. Parvinen, On the mechanistic underpinning of discrete-time population models, Theor. Popul. Biol., 72 (2007), 41-51. |
[15] |
M. S. Fowler and G. D. Ruxton, Population dynamic consequences of Allee effects, J. Theor. Biol., 215 (2002), 39-46.
doi: 10.1006/jtbi.2001.2486. |
[16] |
A. Friedman and A.-A. Yakubu, Fatal disease and demographic allee effect, population persistence and extinction, J. Biological Dynamics, 6 (2012), 495-508.
doi: 10.1080/17513758.2011.630489. |
[17] |
G. R. J. Gaut, K. Goldring, F. Grogan, C. Haskell and R. J. Sacker, Difference equations with the Allee effect and the periodic sigmoid Beverton-Holt equation revisited, J. Biological Dynamics, 6 (2012), 1019-1033.
doi: 10.1080/17513758.2012.719039. |
[18] |
A. J. Harry, C. M. Kent and V. L. Kocic, Global behavior of solutions of a periodically forced Sigmoid Beverton-Holt model, J. Biological Dynamics, 6 (2012), 212-234.
doi: 10.1080/17513758.2011.552738. |
[19] |
S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model, J. Difference Eq. & Appl., 12 (2006), 165-181.
doi: 10.1080/10236190500539238. |
[20] |
V. L. Kocic, A note on the nonautonomous Beverton-Holt model, J. Difference Eq. and Appl., 11 (2005), 415-422.
doi: 10.1080/10236190412331335463. |
[21] |
R. Kon, A note on attenuant cycles of population models with periodic carrying capacity, J. Difference Eq. Appl., 10 (2004), 791-793.
doi: 10.1080/10236190410001703949. |
[22] |
R. Kon, Attenuant cycles of population models with periodic carrying capacity, J. Difference Eq. Appl., 11 (2005), 423-430.
doi: 10.1080/10236190412331335472. |
[23] |
J. Li, B. Song and X. Wang, An extended discrete Ricker population model with Allee effects, J. Difference Eq. & Appl, 13 (2007), 309-321.
doi: 10.1080/10236190601079191. |
[24] |
R. Luís, E. Elaydi and H. Oliveira, Nonautonomous periodic systems with Allee effects, J. Difference Eq. & Appl., 16 (2010), 1179-1196.
doi: 10.1080/10236190902794951. |
[25] |
R. A. Myers, N. J. Barrowman, J. A. Hutchings and A. A. Rosenberg, Population dynamics of exploited fish stocks at low population levels, Science, 269 (1995), 1106-1108.
doi: 10.1126/science.269.5227.1106. |
[26] |
C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics, 2002, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-14258-1. |
[27] |
R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.
doi: 10.1080/10236190601008752. |
[28] |
S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Pop. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[29] |
P. A. Stephens and W. J. Sutherland, Vertebrate mating systems, Allee effects and conservation, In M. Apollonio, M. Festa-Bianchet, and D. Mainardi, editors, Vertebrate mating systems, pages 186-213, Singapore, 2000. World Scientific Publishing.
doi: 10.1142/9789812793584_0009. |
[30] |
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect? Oikos, 87 (1999), 185-190.
doi: 10.2307/3547011. |
[31] |
C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecology Letters, 8 (2005), 895-908.
doi: 10.1111/j.1461-0248.2005.00787.x. |
[32] |
A.-A. Yakubu, Allee effects in discrete-time SIUS epidemic models with infected newborns, J. Difference Eq. & Appl., 13 (2007), 341-356.
doi: 10.1080/10236190601079076. |
[33] |
Y. Yang and R. J. Sacker, Resonance and attenuation in the $n$-periodic Beverton-Holt Equation, J. Difference Eq & Appl., 19 (2013), 1174-1191. |
[34] |
S. Zhou, Y. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23-31.
doi: 10.1016/j.tpb.2004.06.007. |
show all references
References:
[1] |
L. Allen, J. Fagan, G. Hognas and H. Fagerholm, Population extinction in discrete-time stochastic population models with an Allee effect, J. Difference Eq. & Appl., 11 (2005), 273-293.
doi: 10.1080/10236190412331335373. |
[2] |
R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, volume 11 of Fish and Fisheries Series, Chapman and Hall, London, 1957, reprinted, 1993. |
[3] |
J. M. Cushing, The Allee effect in age-structured population dynamics, In T. Hallam, L. Gross, and S. Levin, editors, Mathematical Ecology, pages 479-505. World Sci. Publ., Teaneck, NJ, 1988. |
[4] |
J. M. Cushing, Oscillations in age-structured population models with Allee effect, J. Comput. and Appl. Math., 52 (1994), 71-80.
doi: 10.1016/0377-0427(94)90349-2. |
[5] |
J. M. Cushing and S. M. Henson, Global dynamics of some periodically forced, monotone difference equations, J. Difference Eq. & Appl., 7 (2001), 857-872.
doi: 10.1080/10236190108808308. |
[6] |
J. M. Cushing and S. M. Henson, A periodically forced Beverton-Holt equation, J. Difference Eq. & Appl., 8 (2002), 1119-1120.
doi: 10.1080/1023619021000053980. |
[7] |
H. G. Davis, C. M. Taylor, J. C. Civille and D. R. Strong, An Allee effect at the front of a plant invasion: Spartina in a Pacific estuary, J. Ecology, 92 (2004), 321-327.
doi: 10.1111/j.0022-0477.2004.00873.x. |
[8] |
B. Dennis, Allee effects: Population growth, critical density and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538. |
[9] |
S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005. |
[10] |
S. Elaydi and R. J. Sacker, Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjecture, In S. Elaydi, G. Ladas, B. Aulbach, and O. Dosly, editors, 8th Internat. Conf. on Difference Equations and Appl.(2003), Brno, Czech Republic, pages 113-126. Chapman and Hall, 2005.
doi: 10.1201/9781420034905. |
[11] |
S. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Difference Eq. & Appl., 11 (2005), 337-346.
doi: 10.1080/10236190412331335418. |
[12] |
S. Elaydi and R. J. Sacker, Periodic difference equations, population biology and the Cushing-Henson conjectures, Math. Biosciences, 201 (2006), 195-207.
doi: 10.1016/j.mbs.2005.12.021. |
[13] |
S. Elaydi and R. J. Sacker, Population models with Allee effect, A new model, J. Biological Dynamics, 4 (2010), 397-408.
doi: 10.1080/17513750903377434. |
[14] |
H. Eskola and K. Parvinen, On the mechanistic underpinning of discrete-time population models, Theor. Popul. Biol., 72 (2007), 41-51. |
[15] |
M. S. Fowler and G. D. Ruxton, Population dynamic consequences of Allee effects, J. Theor. Biol., 215 (2002), 39-46.
doi: 10.1006/jtbi.2001.2486. |
[16] |
A. Friedman and A.-A. Yakubu, Fatal disease and demographic allee effect, population persistence and extinction, J. Biological Dynamics, 6 (2012), 495-508.
doi: 10.1080/17513758.2011.630489. |
[17] |
G. R. J. Gaut, K. Goldring, F. Grogan, C. Haskell and R. J. Sacker, Difference equations with the Allee effect and the periodic sigmoid Beverton-Holt equation revisited, J. Biological Dynamics, 6 (2012), 1019-1033.
doi: 10.1080/17513758.2012.719039. |
[18] |
A. J. Harry, C. M. Kent and V. L. Kocic, Global behavior of solutions of a periodically forced Sigmoid Beverton-Holt model, J. Biological Dynamics, 6 (2012), 212-234.
doi: 10.1080/17513758.2011.552738. |
[19] |
S. R.-J. Jang, Allee effects in a discrete-time host-parasitoid model, J. Difference Eq. & Appl., 12 (2006), 165-181.
doi: 10.1080/10236190500539238. |
[20] |
V. L. Kocic, A note on the nonautonomous Beverton-Holt model, J. Difference Eq. and Appl., 11 (2005), 415-422.
doi: 10.1080/10236190412331335463. |
[21] |
R. Kon, A note on attenuant cycles of population models with periodic carrying capacity, J. Difference Eq. Appl., 10 (2004), 791-793.
doi: 10.1080/10236190410001703949. |
[22] |
R. Kon, Attenuant cycles of population models with periodic carrying capacity, J. Difference Eq. Appl., 11 (2005), 423-430.
doi: 10.1080/10236190412331335472. |
[23] |
J. Li, B. Song and X. Wang, An extended discrete Ricker population model with Allee effects, J. Difference Eq. & Appl, 13 (2007), 309-321.
doi: 10.1080/10236190601079191. |
[24] |
R. Luís, E. Elaydi and H. Oliveira, Nonautonomous periodic systems with Allee effects, J. Difference Eq. & Appl., 16 (2010), 1179-1196.
doi: 10.1080/10236190902794951. |
[25] |
R. A. Myers, N. J. Barrowman, J. A. Hutchings and A. A. Rosenberg, Population dynamics of exploited fish stocks at low population levels, Science, 269 (1995), 1106-1108.
doi: 10.1126/science.269.5227.1106. |
[26] |
C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics, 2002, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-14258-1. |
[27] |
R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92.
doi: 10.1080/10236190601008752. |
[28] |
S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Pop. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[29] |
P. A. Stephens and W. J. Sutherland, Vertebrate mating systems, Allee effects and conservation, In M. Apollonio, M. Festa-Bianchet, and D. Mainardi, editors, Vertebrate mating systems, pages 186-213, Singapore, 2000. World Scientific Publishing.
doi: 10.1142/9789812793584_0009. |
[30] |
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect? Oikos, 87 (1999), 185-190.
doi: 10.2307/3547011. |
[31] |
C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecology Letters, 8 (2005), 895-908.
doi: 10.1111/j.1461-0248.2005.00787.x. |
[32] |
A.-A. Yakubu, Allee effects in discrete-time SIUS epidemic models with infected newborns, J. Difference Eq. & Appl., 13 (2007), 341-356.
doi: 10.1080/10236190601079076. |
[33] |
Y. Yang and R. J. Sacker, Resonance and attenuation in the $n$-periodic Beverton-Holt Equation, J. Difference Eq & Appl., 19 (2013), 1174-1191. |
[34] |
S. Zhou, Y. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theor. Popul. Biol., 67 (2005), 23-31.
doi: 10.1016/j.tpb.2004.06.007. |
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