Nonlinear Dynamics

, Volume 95, Issue 3, pp 1731–1745 | Cite as

Rich spatial–temporal dynamics in a diffusive population model for pioneer–climax species

  • Ying Su
  • Xingfu ZouEmail author
Original Paper


A general diffusive population model for interactions of pioneer and climax species subject to the no-flux boundary condition is considered. Local and global steady-state bifurcations as well as Hopf bifurcations are investigated. A condition for Turing instability not to happen is obtained, and the conditions for occurrences of Turing bifurcations and Hopf bifurcations are also obtained. Numerical simulations are carried out to demonstrate and extend the obtained analytic results which suggest that the spatial diffusion may make the climax species more dominant. The results indicate that the model, with spatial diffusion incorporated , can have very rich spatial–temporal dynamics.


Pioneer species Climax species Diffusion Hopf bifurcation Turing bifurcation Spatial–temporal pattern 

Mathematics Subject Classification

35B32 35K57 92B05 92D25 


  1. 1.
    Allee, W.C.: Animal Aggregations: A Study in General Sociology. University of Chicago Press, Chicago (1931)CrossRefGoogle Scholar
  2. 2.
    Brown, S., Dockery, J., Pernarowski, M.: Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species. Math. Biosci. 194, 21–36 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Buchanan, J.R.: Asymptotic behavior of two interacting pioneer/climax species. Fields Inst. Commun. 21, 51–63 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Buchanan, J.R.: Turing instability in pioneer/climax species interactions. Math. Biosci. 194, 199–216 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction–Diffusion Equations, Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (2003)zbMATHGoogle Scholar
  6. 6.
    Dockery, J., Hutson, V., Mischaikow, K., Pernarowski, M.: The evolution of slow dispersal rates: a reaction–diffusion model. J. Math. Biol. 37, 61–83 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)CrossRefGoogle Scholar
  8. 8.
    Jin, J., Shi, J., Wei, J., Yi, F.: Bifurcations of patterned solutions in the diffusive Lengyel–Epstein system of CIMA chemical reactions. Rocky Mt. J. Math. 43, 1637–1674 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, X., Jiang, W., Shi, J.: Hopf bifurcation and turing instability in the reaction–diffusion Holling–Tanner predator–prey model. IMA J. Appl. Math. 78, 287–300 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Liu, J., Wei, J.: Bifurcation analysis of a diffusive model of pioneer and climax species interaction. Adv. Differ. Equ. 52, 1–11 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Mizoguchi, N., Ninomiya, H., Yanagida, E.: Diffusion-induced blowup in a nonlinear parabolic system. J. Dyn. Differ. Equ. 4, 619–638 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum, New York (1992)zbMATHGoogle Scholar
  13. 13.
    Peng, R., Shi, J., Wang, M.: On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law. Nonlinearity 21, 1471–1488 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Peng, R., Yi, F., Zhao, X.: Spatiotemporal patterns in a reaction–diffusion model with the Degn–Harrison reaction scheme. J. Differ. Equ. 254, 2465–2498 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ricker, W.E.: Stock and recruitment. J. Fish. Res. Board Can. 11, 559–623 (1954)CrossRefGoogle Scholar
  16. 16.
    Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10, 863–874 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Selgrade, J.F., Namkoong, G.: Stable periodic behavior in a pioneer-climax model. Nat. Resour. Model. 4, 215 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Selgrade, J.F., Roberds, J.H.: Lumped-density population models of pioneer-climax type and stability analysis of Hopf bifurcations. Math. Biosci. 135, 1–21 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Song, Y., Zou, X.: Bifurcation analysis of a diffusive ratio-dependent predator–prey model. Nonlinear Dyn. 78, 49–70 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Su, Y., Zou, X.: Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition. Nonlinearity 27, 87–104 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sumner, S.: Competing species models for pioneer-climax forest dynamical systems. Proc. Dyn. Syst. 1, 351–358 (1994)zbMATHGoogle Scholar
  22. 22.
    Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Verhulst, P.F.: Notice sur la loi que la population poursuit dans son accroissement. Corresp. Math. Phys. 10, 113–121 (1838)Google Scholar
  24. 24.
    Wang, J., Wei, J., Shi, J.: Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems. J. Differ. Equ. 260, 3495–3523 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Weng, P., Zou, X.: Minimal wave speed and spread speed of competing pioneer and climax species. Appl. Anal. 93, 2093–2110 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system. J. Differ. Equ. 246, 1944–1977 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yuan, Z., Zou, X.: Co-invasion waves in a reaction diffusion model for competing pioneer and climax species. Nonlinear Anal. RWA 11, 232–245 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhou, J., Shi, J.: Pattern formation in a general glycolysis reaction difusion system IMA. J. Appl. Math. 80, 1703–1738 (2015)zbMATHGoogle Scholar
  29. 29.
    Zuo, W., Wei, J.: Multiple bifurcations and spatiotemporal patterns for a coupled two-cell Brusselator model. Dyn. Partial Differ. Equ. 8, 363–384 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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