Skip to main content
SpringerLink
Log in

Population dispersal and Allee effect

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

This paper studies influences of population dispersal on the dynamics of populations that live in patches and grow under Allee effect. Analytical conditions for the global stability of the model in the case of weak Allee effect are established by using the theory of monotonic dynamical systems. Numerical simulations are provided for the case of two patches and strong Allee effect, which reveal that a moderate migration to the better patch is beneficial to overall population, whereas a larger one is harmful.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Amarasekare, P.: Interactions between local dynamics and dispersal: insights from single species models. Theor. Popul. Biol. 53, 44–59 (1998)

    Article  MATH  Google Scholar 

  2. Arino, J., van den Driessche, P.: A multi-city epidemic model. Math. Popul. Stud. 10, 175–193 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berman, A., Plemmon, R.J.: Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, vol. 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)

  4. Berec, L.: Models of Allee effects and their implications for population and community dynamics. Biophys. Rev. Lett. 3, 157–181 (2008)

    Article  Google Scholar 

  5. Boukal, D.S., Berec, L.: Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. J. Theor. Biol. 218, 375–394 (2002)

    Article  MathSciNet  Google Scholar 

  6. Courchamp, F., Berec, L., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press, Oxford (2008)

    Book  Google Scholar 

  7. Cushing, J.M.: The evolutionary dynamics of a population model with a strong Allee effect. Math. Biosci. 12, 643–660 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dennis, B.: Allee effects: population growth, critical density, and the chance of extinction. Nat. Resour. Model. 3, 481–538 (1989)

    MathSciNet  MATH  Google Scholar 

  9. Dennis, B.: Allee effects in stochastic populations. Oikos 96, 389–401 (2002)

    Article  Google Scholar 

  10. Du, Y., Shi, J.: Allee effect and bistability in a spatially heterogeneous predator–prey model. Trans. Am. Math. Soc. 359, 4557–4593 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, D., Ruan, S.: A multipatch malaria model with logistic growth populations. SIAM J. Appl. Math. 72, 819–841 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hirsch, M.W.: Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383, 1–53 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Kang, Y., Lanchier, N.: Expansion or extinction: deterministic and stochastic two-patch models with Allee effects. J. Math. Biol. 62, 925–973 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Keitt, T.H., Lewis, M.A., Holt, R.D.: Allee effects, invasion pinning, and species borders. Am. Nat. 157, 203–216 (2001)

    Article  Google Scholar 

  16. Kramer, A.M., Dennis, B., Liebhold, A.M., Drake, J.M.: The evidence for Allee effects. Popul. Ecol. 51, 341–354 (2009)

    Article  Google Scholar 

  17. Levin, S.A.: Dispersion and population interactions. Am. Nat. 108, 207–228 (1974)

    Article  Google Scholar 

  18. McCarthy, M.A.: The Allee effect, finding mates and theoretical models. Ecol. Model. 103, 99–102 (1997)

    Article  Google Scholar 

  19. Padrón, V., Trevisan, M.C.: Effect of aggregating behavior on population recovery on a set of habitat islands. Math. Biosci. 165, 63–78 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Smith, H.L.: Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41. American Mathematical Society, Providence (1995)

  21. Takeuchi, Y.: Diffusion-mediated persistence in two-species competition Lotka–Volterra model. Math. Biosci. 95, 65–83 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Takeuchi, Y., Lu, Z.: Permanence and global stability for competitive Lotka–Volterra diffusion systems. Nonlinear Anal. 24, 91–104 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Terry, A.J.: Predator–prey models with component Allee effect for predator reproduction. J. Math. Biol. 71, 1325–1352 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, W., Fergola, P., Tenneriello, C.: Global attractivity of periodic solutions of population models. J. Math. Anal. Appl. 211, 498–511 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, W., Fergola, P., Tenneriello, C.: Innovation diffusion model in patch environment. Appl. Math. Comput. 134, 51–67 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Wang, W., Zhao, X.: An epidemic model in a patchy environment. Math. Biosci. 190, 97–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, W., Zhao, X.: An age-structured epidemic model in a patchy environment. SIAM J. Appl. Math. 65, 1597–1614 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, W., Zhao, X.: An epidemic model with population dispersal and infection period. SIAM J. Appl. Math. 66, 1454–1472 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, W., Ma, X.: Harmless delays for uniform persistence. J. Math. Anal. Appl. 158, 256–268 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, W., Takeuchi, Y.: Adaptation of prey and predators between patches. J. Theor. Biol. 258, 603–613 (2009)

    Article  MathSciNet  Google Scholar 

  31. Xu, D., Feng, Z., Linda, J., S., Allen, Swihart, R.K.: A spatially structured metapopulation model with patch dynamics. J. Theor. Biol. 239, 469–481 (2006)

  32. Zhang, X., Wang, W.: Influences of migrations from local competitive pressures on populations between patches. J. Appl. Math. Comput. 37, 313–330 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhao, X.Q.: Dynamical Systems in Population Biology. Springer, New York (2003)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Wendi Wang

Authors
  1. Wendi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wendi Wang.

Additional information

This work was supported by the National Natural Science Found of China (11571284).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, W. Population dispersal and Allee effect. Ricerche mat 65, 535–548 (2016). https://doi.org/10.1007/s11587-016-0273-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-016-0273-0

Keywords

Mathematics Subject Classification

Search

Navigation