March  2014, 19(2): 589-606. doi: 10.3934/dcdsb.2014.19.589

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

Received  April 2013 Revised  October 2013 Published  February 2014

The $\lambda$-Ricker equation has, for certain values of the parameters, an unstable fixed point giving rise to the Allee effect, and an attracting fixed point, the carrying capacity. The $k$-periodic $\lambda$-Ricker equation is studied and parameter intervals are determined for which there exist a $k$-periodic Allee state and a $k$-periodic attracting state.
Citation: Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589
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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. Dennis, Allee effects: Population growth, critical density and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.  Google Scholar

[9]

S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Eskola and K. Parvinen, On the mechanistic underpinning of discrete-time population models, Theor. Popul. Biol., 72 (2007), 41-51. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Kon, Attenuant cycles of population models with periodic carrying capacity, J. Difference Eq. Appl., 11 (2005), 423-430. doi: 10.1080/10236190412331335472.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92. doi: 10.1080/10236190601008752.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. Dennis, Allee effects: Population growth, critical density and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.  Google Scholar

[9]

S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, USA, third edition, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Eskola and K. Parvinen, On the mechanistic underpinning of discrete-time population models, Theor. Popul. Biol., 72 (2007), 41-51. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Kon, Attenuant cycles of population models with periodic carrying capacity, J. Difference Eq. Appl., 11 (2005), 423-430. doi: 10.1080/10236190412331335472.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. J. Sacker, A Note on periodic Ricker maps, J. Difference Eq. & Appl., 13 (2007), 89-92. doi: 10.1080/10236190601008752.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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