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Journal of Mathematical Biology

, Volume 74, Issue 6, pp 1425–1482 | Cite as

An impulsive modelling framework of fire occurrence in a size-structured model of tree–grass interactions for savanna ecosystems

  • V. Yatat
  • P. Couteron
  • J. J. Tewa
  • S. Bowong
  • Y. Dumont
Article

Abstract

Fires and mean annual rainfall are major factors that regulate woody and grassy biomasses in savanna ecosystems. Within the savanna biome, conditions of long-lasting coexistence of trees and grasses have been often studied using continuous-time modelling of tree–grass competition. In these studies, fire is a time-continuous forcing while the relationship between woody plant size and fire-sensitivity is not systematically considered. In this paper, we propose a new mathematical framework to model tree–grass interactions that takes into account both the impulsive nature of fire occurrence and size-dependent fire sensitivity (via two classes of woody plants). We carry out a qualitative analysis that highlights ecological thresholds and bifurcation parameters that shape the dynamics of the savanna-like systems within the main ecological zones. Through a qualitative analysis, we show that the impulsive modelling of fire occurrences leads to more diverse behaviors including cases of grassland, savanna and forest tristability and a more realistic array of solutions than the analogous time-continuous fire models. Numerical simulations are carried out with respect to the three main ecological contexts (moist, mesic, semi-arid) to illustrate the theoretical results and to support a discussion about the bifurcation parameters and the advantages of the model.

Keywords

Savanna Fire Asymmetric competition Impulsive differential equation Qualitative analysis Nonstandard finite difference scheme 

Mathematics Subject Classification

34K45 34K28 92D40 

Notes

Acknowledgments

Thanks to the reviewers for insightful comments that helped us to improve the paper. The first author is grateful to the French government and the SCAC service of the French Embassy in Yaoundé (Cameroon) for their support (SCAC fund) during the preparation of this manuscript.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • V. Yatat
    • 1
    • 2
    • 3
  • P. Couteron
    • 4
    • 5
  • J. J. Tewa
    • 1
    • 2
    • 3
  • S. Bowong
    • 2
    • 3
    • 6
  • Y. Dumont
    • 7
  1. 1.University of Yaoundé I, LIRIMA, GRIMCAPE teamYaoundéCameroon
  2. 2.African Center of Excellence in Information and Communication TechnologiesUniversity of Yaoundé IYaoundéCameroon
  3. 3.IRD, UMI 209, UMMISCO, IRD France NordBondyFrance
  4. 4.IRD, UMR AMAPMontpellierFrance
  5. 5.Plant Systematic and Ecology Laboratory, Higher Teachers’ Training CollegeUniversity of Yaoundé IYaoundéCameroon
  6. 6.University of Douala, LIRIMA, GRIMCAPE teamDoualaCameroon
  7. 7.CIRAD, UMR AMAPMontpellierFrance

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