Nonlinear Dynamics

, Volume 85, Issue 1, pp 1–12 | Cite as

Mathematical modeling of population dynamics with Allee effect

  • Gui-Quan Sun


Allee effect that refers to a positive relationship between individual fitness and population density provides an important conceptual framework in conservation biology. While declining Allee effect causes reduction in extinction risk in low-density population, it provides a benefit in limiting establishment success or spread of invading species. Population models that incorporated Allee effect confer the fundamental role which plays for shaping the population dynamics. In particular, non-spatial predator–prey and invasion models have shown the influence of Allee effects on population dynamics, and spatial models have illustrated its critical roles for pattern formation and the manifestation of traveling wave fronts. We highlight all such no-spatial and spatial population models and their contributions in deeper understanding of population dynamics. In addition, we briefly outline the trends for future research on Allee effect which we think are interesting and widely open.


Allee effect Predator–prey model Diffusion Spatial pattern Noise 



The project is supported by the National Natural Science Foundation of China under Grant 11301490, 131 Talents of Shanxi University, Program for the Outstanding Innovative Teams (OIT) of Higher Learning Institutions of Shanxi and International Exchange Program of Postdoctor in Fudan University.


  1. 1.
    Ackleh, A.S., Allen, L.J., Carter, J.: Establishing a beachhead: a stochastic population model with an allee effect applied to species invasion. Theor. Popul. Biol. 71, 290–300 (2007)CrossRefMATHGoogle Scholar
  2. 2.
    Allee, W.C.: Animal Aggregations. University of Chicago Press, Chicago (1931)Google Scholar
  3. 3.
    Allee, W.C.: Cooperation Among Animals. Henry Shuman, New York (1951)Google Scholar
  4. 4.
    Allee, W.C.: The Social Life of Animals. Beacon Press, Boston (1958)Google Scholar
  5. 5.
    Allee, W.C., Bowen, E.: Studies in animal aggregations: mass protection against colloidal silver among goldfishes. J. Exp. Zool. 61, 185–207 (1932)CrossRefGoogle Scholar
  6. 6.
    Allee, W.C., Emerson, O., Park, T., Schmidt, K.: Principles of Animal Ecology. Saunders, Philadelphia (1949)Google Scholar
  7. 7.
    Almeida, R.C., Delphim, S.A., Costa, M.I.D.S.: A numerical model to solve single-species invasion problems with allee effects. Ecol. Model. 192, 601–617 (2006)CrossRefGoogle Scholar
  8. 8.
    Andrewartha, H.G., Birch, L.C.: The Distribution and Abundance of Animals. University of Chicago Press, Chicago (1954)Google Scholar
  9. 9.
    Assaf, M., Meerson, B.: Extinction of metastable stochastic populations. Phys. Rev. E 81, 021116 (2010)CrossRefGoogle Scholar
  10. 10.
    Bashkirtseva, I., Ryashko, L.: Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with allee effect. Chaos 21, 047514 (2011)CrossRefMATHGoogle Scholar
  11. 11.
    Baurmann, M., Gross, T., Feudel, U.: Instabilities in spatially extended predator–prey systems: spatio-temporal patterns in the neighborhood of turing–hopf bifurcations. J. Theor. Biol. 245, 220–229 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berec, L., Angulo, E., Courchamp, F.: Multiple allee effects and population management. Trends Ecol. Evol. 22, 185–191 (2007)CrossRefGoogle Scholar
  13. 13.
    Bertram, B.C.R.: Living in Groups: Predators and Prey. Blackwell Scientific, Oxford (1978)Google Scholar
  14. 14.
    Bessa-Gomes, C., Legendre, S., Clobert, J.: Allee effects, mating systems and the extinction risk in populations with two sexes. Ecol. Lett. 7, 802–812 (2004)CrossRefGoogle Scholar
  15. 15.
    Birkhead, T.R.: The effects of habitat and density on breeding success in the common guillemot (uria aalge). J. Anim. Ecol. 46, 751–764 (1977)CrossRefGoogle Scholar
  16. 16.
    Boukal, D.S., Sabelis, M.W., Berec, L.: How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor. Popul. Biol. 72, 136–147 (2007)CrossRefMATHGoogle Scholar
  17. 17.
    Chen, J.-X., Peng, L., Zheng, Q., Zhao, Y.-H., Ying, H.-P.: Influences of periodic mechanical deformation on pinned spiral waves. Chaos 24, 033103 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Chen, J.-X., Peng, L., Ma, J., Ying, H.-P.: Liberation of a pinned spiral wave by a rotating electric pulse. EPL 107, 38001 (2014)CrossRefGoogle Scholar
  19. 19.
    Clark, C.W.: Possible effects of schooling on the dynamics of exploited fish populations. J. Cons. Int. Explor. Mer 36, 7–14 (1974)CrossRefGoogle Scholar
  20. 20.
    Compte, A.: Stochastic foundations of fractional dynamics. Phys. Rev. E 53, 4191–4193 (1996)CrossRefGoogle Scholar
  21. 21.
    Courchamp, F., Berec, J., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press, Oxford (2008)CrossRefGoogle Scholar
  22. 22.
    Courchamp, F., Clutton-Brock, T., Grenfell, B.: Inverse density dependence and the allee effect. Trends Ecol. Evol. 14, 405–410 (1999)CrossRefGoogle Scholar
  23. 23.
    Courchamp, F., Grenfell, B.T., Clutton-Brock, T.H.: Impact of natural enemies on obligately cooperative breeders. Oikos 91, 311–322 (2000)CrossRefGoogle Scholar
  24. 24.
    Cushing, J., Hudson, J.T.: Evolutionary dynamics and strong allee effects. J. Biol. Dyn. 6, 941–958 (2012)CrossRefGoogle Scholar
  25. 25.
    Dennis, B.: Allee-effects: population growth, critical density, and the chance of extinction. Nat. Res. Model. 3, 481–538 (1989)MathSciNetMATHGoogle Scholar
  26. 26.
    Dennis, B.: Allee effects in stochastic populations. Oikos 96, 389–401 (2002)CrossRefGoogle Scholar
  27. 27.
    Deredec, A., Courchmp, F.: Extinction thresholds in host-parasite dynamics. Ann. Zool. Fenn. 40, 115–130 (2003)Google Scholar
  28. 28.
    Elaydi, S.N., Sacker, R.J.: Population models with allee effect: a new model. J. Biol. Dyn. 4, 397–408 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Fasani, S., Rinaldi, S.: Remarks on cannibalism and pattern formation in spatially extended prey–predator systems. Nonlinear Dyn. 62, 2543–2548 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Freund, J., Schimansky-Geier, L., Beisner, B., Neiman, A., Russell, D., Yakusheva, T., Moss, F.: Behavioral stochastic resonance: how the noise from a daphnia swarm enhances individual prey capture by juvenile paddlefish. J. Theor. Biol. 214, 71–83 (2002)CrossRefGoogle Scholar
  31. 31.
    Gascoigne, J.C., Lipcius, R.N.: Allee effects driven by predation. J. Appl. Ecol. 41, 801–810 (2004)CrossRefGoogle Scholar
  32. 32.
    Giona, M., Roman, H.: Fractional diffusion equation for transport phenomena in random media. Phys. A 185, 87–97 (1992)CrossRefGoogle Scholar
  33. 33.
    Gourley, S., Kuang, Y.: A stage structured predator–prey model and its dependence on through-stage delay and death rate. J. Math. Biol. 49, 188–200 (2004)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Groom, M.: Allee effects limit population viability of an annual plant. Am. Nat. 151, 487–496 (1998)CrossRefGoogle Scholar
  35. 35.
    Gyllenberg, M., Parvinen, K.: Necessary and sufficient conditions for evolutionary suicide. Bull. Math. Biol. 63, 981–993 (2001)CrossRefMATHGoogle Scholar
  36. 36.
    Henry, B., Wearne, S.: Existence of turing instabilities in a two-species fractional reaction–diffusion system. SIAM J. Appl. Math. 62, 870–887 (2002)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Hilker, F., Langlais, M., Malchow, H.: The allee effect and infectious diseases: extinction, multistability, and the (dis-) appearance of oscillations. Am. Nat. 173, 72–88 (2009)CrossRefGoogle Scholar
  39. 39.
    Hilker, F.M., Langlais, M., Petrovskii, S.V., Malchow, H.: A diffusive SI model with Allee effect and application to FIV. Math. Biosci. 206, 61–80 (2007)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Hilker, F.M., Lewis, M.A., Seno, H., Langlais, M., Malchow, H.: Pathogens can slow down or reverse invasion fronts of their hosts. Biol. Invasions 7, 817–832 (2005)CrossRefGoogle Scholar
  41. 41.
    Jang, S.R.J.: Dynamics of an age-structured population with allee effects and harvesting. J. Biol. Dyn. 4, 409–427 (2010)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Kang, Y., Lanchier, N.: Expansion or extinction: deterministic and stochastic two-patch models with Allee effects. J. Math. Biol. 62, 925–973 (2011)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Kent, A., Doncaster, C.P., Sluckin, T.: Consequences for predators of rescue and allee effects on prey. Ecol. Model. 162, 233–245 (2003)CrossRefGoogle Scholar
  44. 44.
    Kramer, A., Dennis, B., Liebhold, A., Drake, J.: The evidence for allee effects. Popul. Ecol. 51, 341–354 (2009)CrossRefGoogle Scholar
  45. 45.
    Lande, R.: Demographic stochasticity and Allee effect on a scale with isotropic noise. Oikos 83, 353–358 (1998)CrossRefGoogle Scholar
  46. 46.
    Lee, A.M., Sather, B.E., Engen, S.: Demographic stochasticity, Allee effects, and extinction: the influence of mating system and sex ratio. Am. Nat. 177, 301–313 (2011)CrossRefGoogle Scholar
  47. 47.
    Levy, J., Scott, M., Avioli, L.: Mineral homeostasis in neonates of streptozotocin-induced noninsulin-dependent diabetic rats and in their mothers during pregnancy and lactation. Bone 8, 1–6 (1987)CrossRefGoogle Scholar
  48. 48.
    Lewis, M., Kareiva, P.: Allee dynamics and the spread of invading organisms. Theor. Popul. Biol. 43, 141–158 (1993)CrossRefMATHGoogle Scholar
  49. 49.
    Lewis, M., Pacala, S.: Modeling and analysis of stochastic invasion processes. J. Math. Biol. 41, 387–429 (2000)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Li, A.-W.: Impact of noise on pattern formation in a predator–prey model. Nonlinear Dyn. 66, 689–694 (2011)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Li, L., Jin, Z.: Pattern dynamics of a spatial predator–prey model with noise. Nonlinear Dyn. 67, 1737–1744 (2012)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Liu, P.-P., Xue, Y.: Spatiotemporal dynamics of a predator–prey model. Nonlinear Dyn. 69, 71–77 (2012)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Liu, T.-B., Ma, J., Zhao, Q., Tang, J.: Force exerted on the spiral tip by the heterogeneity in an excitable medium. EPL 104, 58005 (2013)CrossRefGoogle Scholar
  54. 54.
    Lou, Q., Chen, J.-X., Zhao, Y.-H., Shen, F.-R., Fu, Y., Wang, L.-L., Liu, Y.: Control of turbulence in heterogeneous excitable media. Phys. Rev. E 85, 026213 (2012)CrossRefGoogle Scholar
  55. 55.
    Ma, J., Liu, Q., Ying, H., Wu, Y.: Emergence of spiral wave induced by defects block. Commun. Nonlinear Sci. Numer. Simul. 18, 1665–1675 (2013)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Ma, J., Wu, X., Chu, R., Zhang, L.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76, 1951–1962 (2014)CrossRefGoogle Scholar
  57. 57.
    Ma, J., Wang, C.N., Jin, W.Y., Wu, Y.: Transition from spiral wave to target wave and other coherent structures in the networks of Hodgkin–Huxley neurons. Appl. Math. Comput. 217, 3844–3852 (2010)MathSciNetMATHGoogle Scholar
  58. 58.
    MacDonald, N.: Time lags in biological models. In: Lecture Notes in Biomathematics. Springer, Berlin (1985)Google Scholar
  59. 59.
    Matsuda, H., Abrams, P.: Timid consumers: self-extinction due to adaptive change in foraging and anti-predator effort. Theor. Popul. Biol. 45, 76–91 (1994)CrossRefMATHGoogle Scholar
  60. 60.
    Medvinsky, A., Petrovskii, S., Tikhonova, I., Malchow, H., Li, B.-L.: Spatio-temporal complexity of plankton and fish dynamics in simple model ecosystems. SIAM Rev. 44, 311–370 (2002)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Mendez, V., Sans, C., Llopis, I., Campos, D.: Extinction conditions for isolated populations with Allee effect. Math. Biosci. 232, 78–86 (2011)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Metzler, R., Glockle, W., Nonnenmacher, T.: Fractional model equation for anomalous diffusion. Phys. A 211, 13–24 (1994)CrossRefGoogle Scholar
  63. 63.
    Mistro, D.C., Rodrigues, L.A.D., Petrovskii, S.: Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong allee effect. Ecol. Complex. 9, 16–32 (2012)CrossRefGoogle Scholar
  64. 64.
    Moller, A.P., Legendre, S.: Allee effect, sexual selection and demographic stochasticity. Oikos 92, 27–34 (2001)CrossRefGoogle Scholar
  65. 65.
    Morozov, A., Li, B.: On the importance of dimensionality of space in models of space-mediated population persistence. Theor. Popul. Biol. 71, 278–289 (2007)CrossRefMATHGoogle Scholar
  66. 66.
    Morozov, A., Petrovskii, S., Li, B.: Bifurcations and chaos in a predator–prey system with the Allee effect. Proc. R. Soc. Lond. B 271, 1407–1414 (2004)CrossRefGoogle Scholar
  67. 67.
    Morozov, A., Petrovskii, S., Li, B.: Spatiotemporal complexity of patchy invasion in a predator–prey system with the Allee effect. J. Theor. Biol. 238, 18–35 (2006)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Murray, J.D.: Mathematical Biology. Springer, New York (2003)MATHGoogle Scholar
  69. 69.
    Ovaskainen, O., Meerson, B.: Stochastic models of population extinction. Trends Ecol. Evol. 11, 643–652 (2010)CrossRefGoogle Scholar
  70. 70.
    Petrovskii, S., Li, B.-L.: An exactly solvable model of population dynamics with density-dependent migrations and the allee effect. Math. Biosci. 186, 79–91 (2003)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Petrovskii, S., Malchow, H., Li, B.-L.: An exact solution of a diffusive predator–prey system. Proc. R. Soc. A 461, 1029–1053 (2005a)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Petrovskii, S., Morozov, A., Li, B.: Regimes of biological invasion in a predator–prey system with the Allee effect. Bull. Math. Biol. 67, 637–661 (2005b)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Petrovskii, S., Morozov, A., Venturino, E.: Allee effect makes possible patchy invasion in a predator–prey system. Ecol. Lett. 5, 345–352 (2002)CrossRefGoogle Scholar
  74. 74.
    Pinto, C.M.A., Machado, J.T.A.: Fractional central pattern generators for bipedal locomotion. Nonlinear Dyn. 62, 27–37 (2010)MathSciNetCrossRefMATHGoogle Scholar
  75. 75.
    Real, L.: The kinetics of functional response. Am. Nat. 111, 289–300 (1977)CrossRefGoogle Scholar
  76. 76.
    Regoes, R.R., Ebert, D., Bonhoeffer, S.: Dose-dependent infection rates of parasites produce the Allee effect in epidemiology. Proc. R. Soc. Lond. B 269, 271–279 (2002)CrossRefGoogle Scholar
  77. 77.
    Reichenbach, T., Mobilia, M., Frey, E.: Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games. Nature 448, 1046–1049 (2007a)CrossRefGoogle Scholar
  78. 78.
    Reichenbach, T., Mobilia, M., Frey, E.: Noise and correlations in a spatial population model with cyclic competition. Phys. Rev. Lett. 99, 238105 (2007b)CrossRefGoogle Scholar
  79. 79.
    Shi, J., Shivaji, R.: Persistence in reaction diffusion models with weak Allee effect. J. Math. Biol. 52, 807–829 (2006)MathSciNetCrossRefMATHGoogle Scholar
  80. 80.
    Stephens, P., Sutherland, W.: Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecol. Evol. 14, 401–405 (1999)CrossRefGoogle Scholar
  81. 81.
    Sun, G.-Q., Jin, Z., Li, L., Liu, Q.-X.: The role of noise in a predator–prey model with Allee effect. J. Biol. Phys. 35, 185–196 (2009)CrossRefGoogle Scholar
  82. 82.
    Sun, G.-Q., Jin, Z., Liu, Q.-X., Li, B.-L.: Rich dynamics in a predator–prey model with both noise and periodic force. BioSystems 100, 14–22 (2010)CrossRefGoogle Scholar
  83. 83.
    Sun, G.-Q., Jin, Z., Li, L., Li, B.-L.: Self-organized wave pattern in a predator–prey model. Nonlinear Dyn. 60, 265–275 (2010)MathSciNetCrossRefMATHGoogle Scholar
  84. 84.
    Sun, G.-Q., Wu, Z.-Y., Wang, Z., Jin, Z.: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn. 83, 811–819 (2016)MathSciNetCrossRefGoogle Scholar
  85. 85.
    Sun, G.-Q., Zhang, G., Jin, Z.: Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dyn. 58, 75–84 (2009)Google Scholar
  86. 86.
    Takimoto, G.: Early warning signals of demographic regime shifts in invading populations. Popul. Ecol. 51, 419–426 (2009)CrossRefGoogle Scholar
  87. 87.
    Verdy, A.: Modulation of predator–prey interactions by the Allee effect. Ecol. Model. 221, 1098–1107 (2010)CrossRefGoogle Scholar
  88. 88.
    Vergni, D., Iannaccone, S., Berti, S., Cencini, M.: Invasions in heterogeneous habitats in the presence of advection. J. Theor. Biol. 301, 141–152 (2012)MathSciNetCrossRefGoogle Scholar
  89. 89.
    Wang, B., Wang, A.-L., Liu, Y.-J., Liu, Z.-H.: Analysis of a spatial predator–prey model with delay. Nonlinear Dyn. 62, 601–608 (2010)MathSciNetCrossRefMATHGoogle Scholar
  90. 90.
    Wang, G., Liang, X.-G., Wang, F.-Z.: The competitive dynamics of populations subject to an Allee effect. Ecol. Model. 124, 183–192 (1999)CrossRefGoogle Scholar
  91. 91.
    Wang, J., Shi, J., Wei, J.: Predator–prey system with strong allee effect in prey. J. Math. Biol. 62, 291–331 (2011a)MathSciNetCrossRefMATHGoogle Scholar
  92. 92.
    Wang, M.-H., Kot, M.: Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 171, 83–97 (2001)MathSciNetCrossRefMATHGoogle Scholar
  93. 93.
    Wang, M.-H., Kot, M., Neubert, M.G.: Integrodifference equations, Allee effects, and invasions. J. Math. Biol. 44, 150–168 (2002)MathSciNetCrossRefMATHGoogle Scholar
  94. 94.
    Wang, W., Liu, H., Li, Z., Guo, Z., Yang, Y.: Invasion dynamics of epidemic with the Allee effect. BioSystems 105, 25–33 (2011b)CrossRefGoogle Scholar
  95. 95.
    Wang, W.-X., Zhang, Y.-B., Liu, C.: Analysis of a discrete-time predator–prey system with Allee effect. Ecol. Complex. 8, 81–85 (2011c)CrossRefGoogle Scholar
  96. 96.
    Zaslavsky, G.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002)MathSciNetCrossRefMATHGoogle Scholar
  97. 97.
    Zhang, X.-Q., Sun, G.-Q., Jin, Z.: Spatial dynamics in a predator–prey model with Beddington–DeAngelis functional response. Phys. Rev. E 85, 021924 (2012)CrossRefGoogle Scholar
  98. 98.
    Zhou, S.-R., Liu, Y.-F., Wang, G.: The stability of predator–prey systems subject to the Allee effects. Theor. Popul. Biol. 67, 23–31 (2005)CrossRefMATHGoogle Scholar
  99. 99.
    Zhou, S.-R., Wang, G.: Allee-like effects in metapopulation dynamics. Math. Biosci. 189, 103–113 (2004)MathSciNetCrossRefMATHGoogle Scholar
  100. 100.
    Zu, J., Mimura, M., Wakano, J.-Y.: The evolution of phenotypic traits in a predator–prey system subject to Allee effect. J. Theor. Biol. 262, 528–543 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Complex Systems Research CenterShanxi UniversityTaiyuanPeople’s Republic of China
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina

Personalised recommendations