September  2016, 21(7): 2321-2336. doi: 10.3934/dcdsb.2016049

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

Received  February 2015 Revised  November 2015 Published  August 2016

We propose a stochastic logistic model with mate limitation and stochastic immigration. Incorporating stochastic immigration into a continuous time Markov chain model, we derive and analyze the associated master equation. By a standard result, there exists a unique globally stable positive stationary distribution. We show that such stationary distribution admits a bimodal profile which implies that a strong Allee effect exists in the stochastic model. Such strong Allee effect disappears and threshold phenomenon emerges as the total population size goes to infinity. Stochasticity vanishes and the model becomes deterministic as the total population size goes to infinity. This implies that there is only one possible fate (either to die out or survive) for a species constrained to a specific community and whether population eventually goes extinct or persists does not depend on initial population density but on a critical inherent constant determined by birth, death and mate limitation. Such a conclusion interprets differently from the classical ordinary differential equation model and thus a paradox on strong Allee effect occurs. Such paradox illustrates the diffusion theory's dilemma.
Citation: Chuang Xu. Strong Allee effect in a stochastic logistic model with mate limitation and stochastic immigration. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2321-2336. doi: 10.3934/dcdsb.2016049
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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