# 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 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.
 [1] Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 [2] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [3] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [4] Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060 [5] Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 [6] Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051 [7] Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 [8] Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629 [9] Xiaoyuan Chang, Junping Shi. Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021242 [10] Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124 [11] Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073 [12] Yuying Liu, Yuxiao Guo, Junjie Wei. Dynamics in a diffusive predator-prey system with stage structure and strong allee effect. Communications on Pure and Applied Analysis, 2020, 19 (2) : 883-910. doi: 10.3934/cpaa.2020040 [13] Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283 [14] Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135 [15] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [16] Kashi Behrstock, Michel Benaïm, Morris W. Hirsch. Smale strategies for network prisoner's dilemma games. Journal of Dynamics and Games, 2015, 2 (2) : 141-155. doi: 10.3934/jdg.2015.2.141 [17] Sharon M. Cameron, Ariel Cintrón-Arias. Prisoner's Dilemma on real social networks: Revisited. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1381-1398. doi: 10.3934/mbe.2013.10.1381 [18] Fırat Evirgen, Sümeyra Uçar, Necati Özdemir, Zakia Hammouch. System response of an alcoholism model under the effect of immigration via non-singular kernel derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2199-2212. doi: 10.3934/dcdss.2020145 [19] J. Leonel Rocha, Abdel-Kaddous Taha, Danièle Fournier-Prunaret. Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3131-3163. doi: 10.3934/dcdsb.2015.20.3131 [20] Alexei Pokrovskii, Dmitrii Rachinskii. Effect of positive feedback on Devil's staircase input-output relationship. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1095-1112. doi: 10.3934/dcdss.2013.6.1095

2020 Impact Factor: 1.327