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Strong Allee effect in a stochastic logistic model with mate limitation and stochastic immigration
1. | Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China |
References:
[1] |
A. S. Ackleh, L. J. S. Allen and J. Carter, Establishing a beachhead: A stochastic population model with an Allee effect applied to species invasion, Theor. Popul. Biol., 71 (2007), 290-300.
doi: 10.1016/j.tpb.2006.12.006. |
[2] |
W. C. Allee, Animal Aggregations, Univ. of Chicago Press, Chicago, 1931.
doi: 10.1086/394281. |
[3] |
W. C. Allee, The Social Life of Animals, W. W. Norton & Company Inc. Publishers, New York, 1938. |
[4] |
P. Amarasekare, Allee effects in metapopulation dynamics, Am. Nat., 152 (1998), 298-302.
doi: 10.1086/286169. |
[5] |
H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, Univ. of Chicago Press, Chicago, 1954. |
[6] |
B. P. Beirne, Biological control attempts by introductions against pest insects in the field in Canada, Canad. Ent., 107 (1975), 225-236.
doi: 10.4039/Ent107225-3. |
[7] |
F. Brauer, Harvesting strategies for population systems, Rocky Mt. J. Math., 9 (1979), 19-26.
doi: 10.1216/RMJ-1979-9-1-19. |
[8] |
J. R. Chazottes, P. Collet and S. Méléard, Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes, Probab. Theory Rel., 164 (2016), 285-332.
doi: 10.1007/s00440-014-0612-6. |
[9] |
F. Courchamp, J. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford Univ. Press, New York, 2008.
doi: 10.1093/acprof:oso/9780198570301.001.0001. |
[10] |
F. Courchamp, B. Grenfell and T. H. Clutton-Brock, Population dynamics of obligate cooperators, Proc. R. Soc. London Ser. B, 266 (1999), 557-564.
doi: 10.1098/rspb.1999.0672. |
[11] |
B. Dennis, Allee effects: Population growth, critical density, and chance of extinction, Nat. Resour. Model., 3 (1989), 381-538. |
[12] |
B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401.
doi: 10.1034/j.1600-0706.2002.960301.x. |
[13] |
J. M. Drake and A. M. Kramer, Allee effects, Nat. Edu. Knowl., 3 (2011), 2. |
[14] |
R. Frankham, Relationship of genetic variation to population size in wildlife-a review, Conserv. Biol., 10 (1996), 1500-1508. |
[15] |
D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A, 188 (1992), 404-425.
doi: 10.1016/0378-4371(92)90283-V. |
[16] |
N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974. |
[17] |
C. Greene and J. A. Stamps, Habitat selection at low population densities, Ecol., 82 (2001), 2091-2100. |
[18] |
B. Griffith, J. M. Scott, J. W. Carpenter and C. Reed, Translocation as a species conservation tool: Status and strategy, Science, 245 (1989), 477-480.
doi: 10.1126/science.245.4917.477. |
[19] |
M. Gyllenberg, J. Hemminki and T. Tammaru, Allee effects can both conserve and create spatial heterogeneity in population densities, Theor. Popul. Biol., 56 (1999), 231-242.
doi: 10.1006/tpbi.1999.1430. |
[20] |
J. B. S. Haldane, Animal populations and their regulation, New Biol., 15 (1953), 9-24. |
[21] |
G. Huberman, Qualitative behavior of a fishery system, Math. Biosci., 42 (1978), 1-14.
doi: 10.1016/0025-5564(78)90002-0. |
[22] |
Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects, J. Math. Biol., 62 (2011), 925-973.
doi: 10.1007/s00285-010-0359-3. |
[23] |
D. G. Kendall, Deterministic and stochastic epidemics in closed populations, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (ed. J. Neyman), Univ. of California Press, 4 (1956), 149-165. |
[24] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects, Popul. Ecol., 51 (2009), 341-354.
doi: 10.1007/s10144-009-0152-6. |
[25] |
R. Lande, Demographic stochasticity and Allee effect on a scale with isotropic noise, Oikos, 83 (1998), 353-358. |
[26] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 43 (1993), 141-158.
doi: 10.1006/tpbi.1993.1007. |
[27] |
D. Ludwig, D. D. Jones and C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol., 47 (1978), 315-332.
doi: 10.2307/3939. |
[28] |
M. A. McCarthy, The Allee effect, finding mates and theoretical models, Ecol. Model., 103 (1997), 99-102.
doi: 10.1016/S0304-3800(97)00104-X. |
[29] |
R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
doi: 10.1038/269471a0. |
[30] |
I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, J. Theor. Biol., 211 (2001), 11-27. |
[31] |
R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations, John Wiley & Sons, New York, 1982. |
[32] |
O. Ovaskainen and B. Meerson, Stochastic models of population extinction, Trends Ecol. Evol., 25 (2010), 643-652.
doi: 10.1016/j.tree.2010.07.009. |
[33] |
J. R. Philip, Sociality and sparse populations, Ecol., 38 (1957), 107-111.
doi: 10.2307/1932132. |
[34] |
H. Qian, Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systems-an analytical theory, Nonlinearity, 24 (2011), R19-R49.
doi: 10.1088/0951-7715/24/6/R01. |
[35] |
I. Scheuring, Allee effect and the dynamical stability of populations, J. Theor. Biol., 199 (1999), 407-414.
doi: 10.1006/jtbi.1999.0966. |
[36] |
S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[37] |
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.
doi: 10.2307/3547011. |
[38] |
C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895-908.
doi: 10.1111/j.1461-0248.2005.00787.x. |
[39] |
N. G. van Kampen, Stochastic Processes in Physics and Chemistry, $3^{rd}$ edition, North-Holland, Amsterdam, 1981.
doi: 10.1063/1.2915501. |
[40] |
A. W. van der Vaart, Asymptotic Statistics, Cambridge Univ. Press, Cambridge, MA, 1998.
doi: 10.1017/CBO9780511802256. |
[41] |
R. R. Veit and M. A. Lewis, Dispersal, population growth, and the Allee effect: Dynamics of the house finch invasion of eastern North America, Am. Nat., 148 (1996), 255-274.
doi: 10.1086/285924. |
[42] |
M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox, Bull. Math. Biol., 69 (2007) , 1727-1746.
doi: 10.1007/s11538-006-9188-3. |
[43] |
D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. |
[44] |
G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci., 11 (1971), 261-265.
doi: 10.1016/0025-5564(71)90087-3. |
[45] |
H. Wells, E. G. Strauss, M. A. Rutter and P. H. Wells, Mate location, population growth, and species extinction, Biol. Conserv., 86 (1998), 317-324.
doi: 10.1016/S0006-3207(98)00032-9. |
[46] |
S. R. Zhou, C. Z. Liu and G. Wang, The competitive dynamics of metapopulation subject to the Allee-like effect, Theor. Popul. Biol., 65 (2004), 29-37.
doi: 10.1016/j.tpb.2003.08.002. |
show all references
References:
[1] |
A. S. Ackleh, L. J. S. Allen and J. Carter, Establishing a beachhead: A stochastic population model with an Allee effect applied to species invasion, Theor. Popul. Biol., 71 (2007), 290-300.
doi: 10.1016/j.tpb.2006.12.006. |
[2] |
W. C. Allee, Animal Aggregations, Univ. of Chicago Press, Chicago, 1931.
doi: 10.1086/394281. |
[3] |
W. C. Allee, The Social Life of Animals, W. W. Norton & Company Inc. Publishers, New York, 1938. |
[4] |
P. Amarasekare, Allee effects in metapopulation dynamics, Am. Nat., 152 (1998), 298-302.
doi: 10.1086/286169. |
[5] |
H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, Univ. of Chicago Press, Chicago, 1954. |
[6] |
B. P. Beirne, Biological control attempts by introductions against pest insects in the field in Canada, Canad. Ent., 107 (1975), 225-236.
doi: 10.4039/Ent107225-3. |
[7] |
F. Brauer, Harvesting strategies for population systems, Rocky Mt. J. Math., 9 (1979), 19-26.
doi: 10.1216/RMJ-1979-9-1-19. |
[8] |
J. R. Chazottes, P. Collet and S. Méléard, Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes, Probab. Theory Rel., 164 (2016), 285-332.
doi: 10.1007/s00440-014-0612-6. |
[9] |
F. Courchamp, J. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford Univ. Press, New York, 2008.
doi: 10.1093/acprof:oso/9780198570301.001.0001. |
[10] |
F. Courchamp, B. Grenfell and T. H. Clutton-Brock, Population dynamics of obligate cooperators, Proc. R. Soc. London Ser. B, 266 (1999), 557-564.
doi: 10.1098/rspb.1999.0672. |
[11] |
B. Dennis, Allee effects: Population growth, critical density, and chance of extinction, Nat. Resour. Model., 3 (1989), 381-538. |
[12] |
B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401.
doi: 10.1034/j.1600-0706.2002.960301.x. |
[13] |
J. M. Drake and A. M. Kramer, Allee effects, Nat. Edu. Knowl., 3 (2011), 2. |
[14] |
R. Frankham, Relationship of genetic variation to population size in wildlife-a review, Conserv. Biol., 10 (1996), 1500-1508. |
[15] |
D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A, 188 (1992), 404-425.
doi: 10.1016/0378-4371(92)90283-V. |
[16] |
N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974. |
[17] |
C. Greene and J. A. Stamps, Habitat selection at low population densities, Ecol., 82 (2001), 2091-2100. |
[18] |
B. Griffith, J. M. Scott, J. W. Carpenter and C. Reed, Translocation as a species conservation tool: Status and strategy, Science, 245 (1989), 477-480.
doi: 10.1126/science.245.4917.477. |
[19] |
M. Gyllenberg, J. Hemminki and T. Tammaru, Allee effects can both conserve and create spatial heterogeneity in population densities, Theor. Popul. Biol., 56 (1999), 231-242.
doi: 10.1006/tpbi.1999.1430. |
[20] |
J. B. S. Haldane, Animal populations and their regulation, New Biol., 15 (1953), 9-24. |
[21] |
G. Huberman, Qualitative behavior of a fishery system, Math. Biosci., 42 (1978), 1-14.
doi: 10.1016/0025-5564(78)90002-0. |
[22] |
Y. Kang and N. Lanchier, Expansion or extinction: Deterministic and stochastic two-patch models with Allee effects, J. Math. Biol., 62 (2011), 925-973.
doi: 10.1007/s00285-010-0359-3. |
[23] |
D. G. Kendall, Deterministic and stochastic epidemics in closed populations, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (ed. J. Neyman), Univ. of California Press, 4 (1956), 149-165. |
[24] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects, Popul. Ecol., 51 (2009), 341-354.
doi: 10.1007/s10144-009-0152-6. |
[25] |
R. Lande, Demographic stochasticity and Allee effect on a scale with isotropic noise, Oikos, 83 (1998), 353-358. |
[26] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theor. Popul. Biol., 43 (1993), 141-158.
doi: 10.1006/tpbi.1993.1007. |
[27] |
D. Ludwig, D. D. Jones and C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol., 47 (1978), 315-332.
doi: 10.2307/3939. |
[28] |
M. A. McCarthy, The Allee effect, finding mates and theoretical models, Ecol. Model., 103 (1997), 99-102.
doi: 10.1016/S0304-3800(97)00104-X. |
[29] |
R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
doi: 10.1038/269471a0. |
[30] |
I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, J. Theor. Biol., 211 (2001), 11-27. |
[31] |
R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations, John Wiley & Sons, New York, 1982. |
[32] |
O. Ovaskainen and B. Meerson, Stochastic models of population extinction, Trends Ecol. Evol., 25 (2010), 643-652.
doi: 10.1016/j.tree.2010.07.009. |
[33] |
J. R. Philip, Sociality and sparse populations, Ecol., 38 (1957), 107-111.
doi: 10.2307/1932132. |
[34] |
H. Qian, Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systems-an analytical theory, Nonlinearity, 24 (2011), R19-R49.
doi: 10.1088/0951-7715/24/6/R01. |
[35] |
I. Scheuring, Allee effect and the dynamical stability of populations, J. Theor. Biol., 199 (1999), 407-414.
doi: 10.1006/jtbi.1999.0966. |
[36] |
S. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.
doi: 10.1016/S0040-5809(03)00072-8. |
[37] |
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.
doi: 10.2307/3547011. |
[38] |
C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895-908.
doi: 10.1111/j.1461-0248.2005.00787.x. |
[39] |
N. G. van Kampen, Stochastic Processes in Physics and Chemistry, $3^{rd}$ edition, North-Holland, Amsterdam, 1981.
doi: 10.1063/1.2915501. |
[40] |
A. W. van der Vaart, Asymptotic Statistics, Cambridge Univ. Press, Cambridge, MA, 1998.
doi: 10.1017/CBO9780511802256. |
[41] |
R. R. Veit and M. A. Lewis, Dispersal, population growth, and the Allee effect: Dynamics of the house finch invasion of eastern North America, Am. Nat., 148 (1996), 255-274.
doi: 10.1086/285924. |
[42] |
M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox, Bull. Math. Biol., 69 (2007) , 1727-1746.
doi: 10.1007/s11538-006-9188-3. |
[43] |
D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. |
[44] |
G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci., 11 (1971), 261-265.
doi: 10.1016/0025-5564(71)90087-3. |
[45] |
H. Wells, E. G. Strauss, M. A. Rutter and P. H. Wells, Mate location, population growth, and species extinction, Biol. Conserv., 86 (1998), 317-324.
doi: 10.1016/S0006-3207(98)00032-9. |
[46] |
S. R. Zhou, C. Z. Liu and G. Wang, The competitive dynamics of metapopulation subject to the Allee-like effect, Theor. Popul. Biol., 65 (2004), 29-37.
doi: 10.1016/j.tpb.2003.08.002. |
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