# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] W. C. Allee, Animal Aggregations, Univ. of Chicago Press, Chicago, 1931. doi: 10.1086/394281.  Google Scholar [3] W. C. Allee, The Social Life of Animals, W. W. Norton & Company Inc. Publishers, New York, 1938. Google Scholar [4] P. Amarasekare, Allee effects in metapopulation dynamics, Am. Nat., 152 (1998), 298-302. doi: 10.1086/286169.  Google Scholar [5] H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, Univ. of Chicago Press, Chicago, 1954. Google Scholar [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.  Google Scholar [7] F. Brauer, Harvesting strategies for population systems, Rocky Mt. J. Math., 9 (1979), 19-26. doi: 10.1216/RMJ-1979-9-1-19.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] B. Dennis, Allee effects: Population growth, critical density, and chance of extinction, Nat. Resour. Model., 3 (1989), 381-538.  Google Scholar [12] B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401. doi: 10.1034/j.1600-0706.2002.960301.x.  Google Scholar [13] J. M. Drake and A. M. Kramer, Allee effects, Nat. Edu. Knowl., 3 (2011), 2. Google Scholar [14] R. Frankham, Relationship of genetic variation to population size in wildlife-a review, Conserv. Biol., 10 (1996), 1500-1508. Google Scholar [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.  Google Scholar [16] N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974.  Google Scholar [17] C. Greene and J. A. Stamps, Habitat selection at low population densities, Ecol., 82 (2001), 2091-2100. Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. B. S. Haldane, Animal populations and their regulation, New Biol., 15 (1953), 9-24. Google Scholar [21] G. Huberman, Qualitative behavior of a fishery system, Math. Biosci., 42 (1978), 1-14. doi: 10.1016/0025-5564(78)90002-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] R. Lande, Demographic stochasticity and Allee effect on a scale with isotropic noise, Oikos, 83 (1998), 353-358. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477. doi: 10.1038/269471a0.  Google Scholar [30] I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, J. Theor. Biol., 211 (2001), 11-27. Google Scholar [31] R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations, John Wiley & Sons, New York, 1982. Google Scholar [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.  Google Scholar [33] J. R. Philip, Sociality and sparse populations, Ecol., 38 (1957), 107-111. doi: 10.2307/1932132.  Google Scholar [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.  Google Scholar [35] I. Scheuring, Allee effect and the dynamical stability of populations, J. Theor. Biol., 199 (1999), 407-414. doi: 10.1006/jtbi.1999.0966.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, $3^{rd}$ edition, North-Holland, Amsterdam, 1981. doi: 10.1063/1.2915501.  Google Scholar [40] A. W. van der Vaart, Asymptotic Statistics, Cambridge Univ. Press, Cambridge, MA, 1998. doi: 10.1017/CBO9780511802256.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] W. C. Allee, Animal Aggregations, Univ. of Chicago Press, Chicago, 1931. doi: 10.1086/394281.  Google Scholar [3] W. C. Allee, The Social Life of Animals, W. W. Norton & Company Inc. Publishers, New York, 1938. Google Scholar [4] P. Amarasekare, Allee effects in metapopulation dynamics, Am. Nat., 152 (1998), 298-302. doi: 10.1086/286169.  Google Scholar [5] H. G. Andrewartha and L. C. Birch, The Distribution and Abundance of Animals, Univ. of Chicago Press, Chicago, 1954. Google Scholar [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.  Google Scholar [7] F. Brauer, Harvesting strategies for population systems, Rocky Mt. J. Math., 9 (1979), 19-26. doi: 10.1216/RMJ-1979-9-1-19.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] B. Dennis, Allee effects: Population growth, critical density, and chance of extinction, Nat. Resour. Model., 3 (1989), 381-538.  Google Scholar [12] B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401. doi: 10.1034/j.1600-0706.2002.960301.x.  Google Scholar [13] J. M. Drake and A. M. Kramer, Allee effects, Nat. Edu. Knowl., 3 (2011), 2. Google Scholar [14] R. Frankham, Relationship of genetic variation to population size in wildlife-a review, Conserv. Biol., 10 (1996), 1500-1508. Google Scholar [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.  Google Scholar [16] N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974.  Google Scholar [17] C. Greene and J. A. Stamps, Habitat selection at low population densities, Ecol., 82 (2001), 2091-2100. Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. B. S. Haldane, Animal populations and their regulation, New Biol., 15 (1953), 9-24. Google Scholar [21] G. Huberman, Qualitative behavior of a fishery system, Math. Biosci., 42 (1978), 1-14. doi: 10.1016/0025-5564(78)90002-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] R. Lande, Demographic stochasticity and Allee effect on a scale with isotropic noise, Oikos, 83 (1998), 353-358. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477. doi: 10.1038/269471a0.  Google Scholar [30] I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, J. Theor. Biol., 211 (2001), 11-27. Google Scholar [31] R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations, John Wiley & Sons, New York, 1982. Google Scholar [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.  Google Scholar [33] J. R. Philip, Sociality and sparse populations, Ecol., 38 (1957), 107-111. doi: 10.2307/1932132.  Google Scholar [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.  Google Scholar [35] I. Scheuring, Allee effect and the dynamical stability of populations, J. Theor. Biol., 199 (1999), 407-414. doi: 10.1006/jtbi.1999.0966.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, $3^{rd}$ edition, North-Holland, Amsterdam, 1981. doi: 10.1063/1.2915501.  Google Scholar [40] A. W. van der Vaart, Asymptotic Statistics, Cambridge Univ. Press, Cambridge, MA, 1998. doi: 10.1017/CBO9780511802256.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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