Journal of Evolution Equations

, Volume 18, Issue 4, pp 1853–1888 | Cite as

A system of state-dependent delay differential equation modeling forest growth I: semiflow properties

  • Pierre MagalEmail author
  • Zhengyang Zhang


In this article, we investigate the semiflow properties of a class of state-dependent delay differential equations which is motivated by some models describing the dynamics of the number of adult trees in forests. We investigate the existence and uniqueness of a semiflow in the space of Lipschitz and \(C^1\) weighted functions. We obtain a blow-up result when the time approaches the maximal time of existence. We conclude the paper with an application of a spatially structured forest model.


State-dependent delay differential equations Forest population dynamics Semiflow Time of blow-up 

Mathematics Subject Classification

34K05 37L99 37N25 


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  1. 1.
    W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM Journal on Applied Mathematics, 52(3) (1992), 855-869.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. F. M. Al-Omari and S. A. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Analysis: Real World Applications, 6 (2005), 13-33.MathSciNetCrossRefGoogle Scholar
  3. 3.
    O. Arino, E. Sanchez, A. Fathallah, State-dependent delay differential equations in population dynamics: modeling and analysis. Topics in Functional Differential and Difference Equations, Fields Institute Communications, Vol. 29, 19–36, American Mathematical Society, 2001.Google Scholar
  4. 4.
    H, Brunner, S. A. Gourley, R. Liu, and Y. Xiao, Pauses of larval development and its consequences for stage-structured populations, SIAM Journal on Applied Mathematics, 77(3) (2017), 977–994.MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, Handbook Of Differential Equations: Ordinary Differential Equations, Vol. 3, 435–545, Elsevier, 2006.Google Scholar
  6. 6.
    M. L. Hbid, M. Louihi and E. Sanchez, A threshold state-dependent delayed functional equation arising from marine population dynamics: modelling and analysis, Journal of Evolution Equations, 10(4) (2010), 905-928.MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Kloosterman, S. A. Campbell and F. J. Poulin, An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton, SIAM Journal on Applied Mathematics, 76(2) (2016), 551-577.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Z. Liu and P. Magal, Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions (in submitted).Google Scholar
  9. 9.
    P. Magal and Z. Zhang, Competition for light in a forest population dynamic model: from computer model to mathematical model, Journal of Theoretical Biology, 419 (2017), 290-304.MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Magal and Z. Zhang, Numerical simulations of a population dynamic model describing parasite destruction in a wild type pine forest, Ecological Complexity (to appear).Google Scholar
  11. 11.
    W. Rudin, Real and Complex Analysis, Third Edition, Tata McGraw-Hill Education, 1987.zbMATHGoogle Scholar
  12. 12.
    H. L. Smith, Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study, Mathematical Biosciences, 113 (1993), 1-23.MathSciNetCrossRefGoogle Scholar
  13. 13.
    H. L. Smith, A structured population model and a related functional differential equation: global attractors and uniform persistence, Journal of Dynamics and Differential Equations, 6(1) (1994), 71-99.MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. L. Smith, Existence and uniqueness of global solutions for a size-structured model of an insect population with variable instar duration, Rocky Mountain Journal of Mathematics, 24(1) (1994), 311-334.MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. L. Smith, Equivalent dynamics for a structured population model and a related functional differential equation, Rocky Mountain Journal of Mathematics, 25(1) (1995), 491-499.MathSciNetCrossRefGoogle Scholar
  16. 16.
    H.-O. Walther, Differential equations with locally bounded delay, Journal of Differential Equations, 252 (2012), 3001-3039.MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.UMR 5251, IMBUniversity of BordeauxBordeauxFrance
  2. 2.UMR 5251, CNRSIMBTalenceFrance

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