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. DumontEmail author


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.


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

Mathematics Subject Classification

34K45 34K28 92D40 



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.


  1. Abbadie L, Gignoux J, Le Roux X, Lepage M (2006) Lamto: structure, functioning, and dynamics of a Savanna Ecosystem. Eco, Stu, Springer, New YorkGoogle Scholar
  2. Accatino F, De Michele C (2013) Humid savanna-forest dynamics: a matrix model with vegetation-fire interactions and seasonality. Ecol Model 265:170–179CrossRefGoogle Scholar
  3. Accatino F, De Michele C, Vezzoli R, Donzelli D, Scholes R (2010) tree-grass co-existence in savanna: interactions of rain and fire. J Theor Biol 267:235–242MathSciNetCrossRefGoogle Scholar
  4. Accatino F, Wiegand K, Ward D, De Michele C (2016) Trees, grass, and fire in humid savannas: the importance of life history traits and spatial processes. Ecol Model 320:135–144CrossRefGoogle Scholar
  5. Anguelov R, Dumont Y, Lubuma JM-S (2012) On nonstandard finite difference schemes in biosciences. AIP Conf Proc 1487:212–223CrossRefGoogle Scholar
  6. Anguelov R, Dumont Y, Lubuma JM-S, Mureithi E (2013) Stability analysis and dynamics preserving non-standar finite difference schemes for malaria model. Math Popul Stud 20(2):101–122MathSciNetCrossRefGoogle Scholar
  7. Anguelov R, Dumont Y, Lubuma JM-S, Shillor M (2014) Dynamically consistent nonstandard finite difference schemes for epidemiological Models. J Comput Appl Math 255:161–182MathSciNetCrossRefzbMATHGoogle Scholar
  8. Archibald S, Roy DP, Van Wilgen B, Scholes RJ (2009) What limits fire? An examination of drivers of burnt area in Southern Africa. Global Change Biol 15:613–630CrossRefGoogle Scholar
  9. Augier P, Lett C, Poggiale JC (2010) Modélisation mathématique en écologie. Cours et exercices corrigés. Dunod, ParisGoogle Scholar
  10. Bainov DD, Simeonov PS (1995) Impulsive differential equations: asymptotic properties of the solutions. World scientific publishing Co, SingaporeCrossRefzbMATHGoogle Scholar
  11. Barbier N, Couteron P, Lefever R, Deblauwe V, Lejeune O (2008) Spatial decoupling of facilitation and competition at the origin of gapped vegetation patterns. Ecology 89:1521–1531CrossRefGoogle Scholar
  12. Baudena M, D’Andrea F, Provenzale A (2010) An idealized model for tree-grass coexistence in savannas: the role of life stage structure and fire disturbances. J Ecol 98:74–80CrossRefGoogle Scholar
  13. Baudena M, Rietkerk M (2013) Complexity and coexistence in a simple spatial model for arid savanna ecosystems. Theor Ecol 6:131–141CrossRefGoogle Scholar
  14. Baudena M, Dekker SC, van Bodegom PM, Cuesta B, Higgins SI, Lehsten V, Reick CH, Rietkerk M, Scheiter S, Yin Z, Zavala MA, Brovkin V (2014) Forest, Savannas and grasslands: bridging the knowledge gap between ecology and dynamic global vegetation models. Biogeosci Discuss 11:9471–9510CrossRefGoogle Scholar
  15. Beckage B, Gross LJ, Platt WJ (2011) Grass feedbacks on fire stabilize savannas. Ecol Model 222:2227–2233CrossRefGoogle Scholar
  16. Belsky AJ (1994) Influuences of trees on savanna productivity: tests of shade, nutrients and tree-grass competition. Ecology 75:922–932CrossRefGoogle Scholar
  17. Belsky AJ, Amundson RG, Duxbury JM, Rika SJ, Ali AR, Mwonga SM (1989) The effects of trees on their physical, chemical, and biological environment in a semi-arid savanna in Kenya. J Appl Ecol 26:1005–1024CrossRefGoogle Scholar
  18. Bond WJ (2008) What limits trees in C4 grasslands and savannas ? Annu Rev Ecol Evol Syst 39:641–59CrossRefGoogle Scholar
  19. Bond WJ, Keeley JE (2005) Fire as a global herbivore: the ecology and evolution of flammable ecosystems. Trends Ecol Evol 20:387–394CrossRefGoogle Scholar
  20. Bond WJ, Midgley GF, Woodward FI (2003) What controls South African vegetation-climate or fire? S Afr J Bot 69:79–91CrossRefGoogle Scholar
  21. Bond WJ, Woodward FI, Midgley GF (2005) The global distribution of ecosystems in a world without fire. New Phytol 165:525–538CrossRefGoogle Scholar
  22. Breman H and Kessler JJ (1995) Woody plants in agroecosystems of semi-arid regions. With an emphasis on the Sahelian countries. Advanced series in Agricultural, vol 23, Springer, BerlinGoogle Scholar
  23. Calabrese JM, Vazquez F, López C, San M, Miguel Grimm V (2010) The independent and interactive effects of tree-tree establishment competition and fire on Savanna structure and dynamics. Am Nat 175:E44–E65CrossRefGoogle Scholar
  24. Chen Y, Liu Z, Haque M (2009) Analysis of a Leslie-Gower-type prey-predator model with periodic impulsive perturbations. Commun Nonlinear Sci Numer Simul 14:3412–3423MathSciNetCrossRefzbMATHGoogle Scholar
  25. Couteron P, Kokou K (1997) Woody vegetation spatial patterns in a semi-arid savanna of Burkina Faso, West Africa. Plant Ecol 132:211–227CrossRefGoogle Scholar
  26. Dai C, Zhao M, Chen L (2012) Dynamic complexity of an Ivlev-type prey-predator system with impulsive state feedback control. J Appl Math 17:1–17 (Article ID 534276)Google Scholar
  27. Deblauwe V, Barbier N, Couteron P, Lejeune O, Bogaert J (2008) The global biogeography of semi-arid periodic vegetation patterns. Global Ecol Biogeogr. doi: 10.1111/j.1466-8238.2008.00413.x Google Scholar
  28. De Michele C, Accatino F, Vezzoli R, Scholes RJ (2011) Savanna domain in the herbivores-fire parameter space exploiting a tree-grass-soil water dynamic model. J Theor Biol 289:74–82CrossRefGoogle Scholar
  29. Diouf A, Barbier N, Lykke AM, Couteron P, Deblauwe V, Mahamane A, Saadou M, Bogaert J (2012) Relationships between fire history, edaphic factor and woody vegetation structure and composition in a semi-arid savanna landscape (Niger, West Africa). Appl Veg Sci 15:488–500CrossRefGoogle Scholar
  30. D’Odorico P, Laio F, Ridolfi LA (2006) probabilistic analysis of fire-induced tree-grass coexistence in savannas. Am Nat 167:E79–E87CrossRefGoogle Scholar
  31. D’Onofrio D (2002) Stability properties of pulse vaccination strategy in SEIR epidemic model. Math Biosci 179:57–72MathSciNetCrossRefzbMATHGoogle Scholar
  32. Dumont Y, Russell JC, Lecomte V, Le Corre M (2010) Conservation of endangered endemic seabirds within a multi-predator context: the Barau’s petrel in Réunion island. Nat Ressour Model 23:381–436CrossRefzbMATHGoogle Scholar
  33. Dumont Y, Tchuenche JM (2012) Mathematical studies on the sterile insect technique for the chikungunya disease and aedes albopictus. J Math Biol 65(5):809–854MathSciNetCrossRefzbMATHGoogle Scholar
  34. Fan M, Kuang Y (2004) Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response. J Math Anal Appl 295:15–39MathSciNetCrossRefzbMATHGoogle Scholar
  35. Favier C, Aleman J, Bremond L, Dubois MA, Freycon V, Yangakola JM (2012) Abrupt shifts in African Savanna tree cover along a climatic gradient. Glob Ecol Biogeogr 21:787–797CrossRefGoogle Scholar
  36. February EC, Higgins SI, Bond WJ, Swemmer L (2013) Influence of competition and rainfall manipulation on the growth responses of savanna trees and grasses. Ecology 94(5):1155–1164CrossRefGoogle Scholar
  37. Fernandez-Oto C, Tlidi M, Escaff D, Clerc MG (2014) Strong interaction between plants induces circular barren patches: fairy circles. Phil Trans R Soc A 372(2027):20140009Google Scholar
  38. Frost PGH, Robertson F (1985) The ecological effects of fire in savannas. In: Walker BH (ed) Determinants of tropical savannas, vol 3., Monograph seriesInternational Council of Scientific Unions Press, Miami, pp 93–140Google Scholar
  39. Gaines RE, Mawhin J (1977) Coincidence degree and nonlinear differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  40. Gignoux J (1994) Modélisation de la coexistence herbes-arbres en savane, PhD ThesisGoogle Scholar
  41. Gignoux J, Lahoreau G, Julliard R, Barot S (2009) Establishment and early persistence of tree seedlings in an annually burned savanna. J Ecol 97:484–495CrossRefGoogle Scholar
  42. Gilad E, Shachak M, Meron E (2007) Dynamics and spatial organization of plant communities in water-limited systems. Theor Popul Biol 72:214–230CrossRefzbMATHGoogle Scholar
  43. Govender N, Trollope WSW, Van Wilgen BW (2006) The effect of fire season, fire frequency, rainfall and management on fire intensity in savanna vegetation in South Africa. J Appl Ecol 43:748–758CrossRefGoogle Scholar
  44. Hale JK (1980) Ordinary differential equations, 2nd edn. Krieger Publishing Company, MalabarzbMATHGoogle Scholar
  45. Hale JK (1988) Asymptotic behavior of dissipative systems. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  46. Higgins SI, Bond WJ, Trollope WSW (2000) Fire, resprouting and variability: a recipe for grass-tree coexistence in savanna. J Ecol 88:213–229CrossRefGoogle Scholar
  47. Higgins SI, Bond WJ, Trollope WSW, Williams RJ (2008) Physically motivated empirical models for the spread and intensity of grass fires. Int J Wildland Fire 17:595–601CrossRefGoogle Scholar
  48. Higgins SI, Shackleton CM, Robinson ER (1999) Changes in woody community structure and composition under constrasting landuse systems in a semi-arid Savanna, South Africa. J Biogeo 26:619–627CrossRefGoogle Scholar
  49. Hirota M, Holmgren M, Van Nes EH, Scheffer M (2011) Global resilience of tropical forest and Savanna to critical transitions. Science 334:232CrossRefGoogle Scholar
  50. Hoffmann WA, Solbrig OT (2003) The role of topkill in the differential response of savanna woody species to fire. For Ecol Manag 180:273–286CrossRefGoogle Scholar
  51. Jeffery KJ, Korte L, Palla F, Walters G, White LJT, Abernethy KA (2014) Fire management in a changing landscape: a case study from Lopé national park. Gabon Parks 20(1):39–52CrossRefGoogle Scholar
  52. Lefever R, Barbier N, Couteron P, Lejeune O (2009) Deeply gapped vegetation patterns: oncrown/root allometry, criticality and desertification. J Theo Ecol 261:194–209Google Scholar
  53. Lehmann C, Prior LD, Williams RJ, Bowman DMJS (2008) Spatio-temporal trends in tree cover of a tropical mesic savanna are driven by landscape disturbance. J Appl Ecol 45:1304–1311CrossRefGoogle Scholar
  54. Maurin O, Davies TJ, Burrows JE, Daru BH, Yessoufou K, Muasya AM, Van der Bank M, Bond JW (2014) Savanna fire and the origins of the underground forests of Africa. New Phytologist 204(1):201–214CrossRefGoogle Scholar
  55. Menaut JC (1983) The vegetation of african savannas. In: Bourlière F (ed) Tropical Savannas, vol 13., Ecosystems of the worldElsevier, Amsterdam, pp 109–149Google Scholar
  56. Menaut JC, César J (1979) Structure and primary productivity of Lamto savannas, Ivory Coast. Ecology 60:1197–1210CrossRefGoogle Scholar
  57. Mermoz S, Réjou-Méchan M, Villard L, Le Toant T, Rossi V, Gourlet-Fleury S (2015) Decrease of L-band SAR backscatter with biomass of dense forests. Rem Sens Environ 159:307–317CrossRefGoogle Scholar
  58. Mermoz S, Le Toant T, Villard L, Réjou-Méchan M, Seifert-Granzin J (2014) Biomass assessment in the Cameroon savanna using ALOS PALSAR data. Rem Sens Environ. doi: 10.1016/j.rse.2014.01.029 Google Scholar
  59. Mordelet P, Menaut JC (1995) Influence of trees on above-ground production dynamics of grasses in a humid savanna. J Veg Sci 6:223–228CrossRefGoogle Scholar
  60. Moro MJ, Pugnaire FI, Haase P, Puigdefabregas J (1997) Effect of the canopy of Retama spaerocarpa on its understorey in a semi-arid environment. Funct Ecol 11:425–431CrossRefGoogle Scholar
  61. Moustakas A, Kunin WE, Cameron TC, Sankaran M (2013) Facilitation or Competition? Tree Effects on Grass Biomass across a Precipitation Gradient. PLoS One 8(2):e57025CrossRefGoogle Scholar
  62. Penning de Vries FWT and Djiteye MA (1982) La productivité des paturages sahéliens. Une étude des sols, des végétations et de l’exploitation de cette ressource naturelle. PUDOC, WageningenGoogle Scholar
  63. Pueyo Y, Kefi S, Alados C, Rietkerk M (2008) Dispersal strategies and spatial organization of vegetation in arid ecosystems. Oikos 117:1522–1532CrossRefGoogle Scholar
  64. Pueyo Y, Kefi S, Daz-Sierra R, Alados C, Rietkerk M (2010) The role of reproductive plant traits and biotic interactions in the dynamics of semi-arid plant communities. Theor Popul Biol 78:289–297CrossRefGoogle Scholar
  65. Sankaran M, Hanan NP, Scholes RJ, Ratnam J, Augustine DJ, Cade BS, Gignoux J, Higgins SI, LeRoux X, Ludwig F, Ardo J, Banyikwa F, Bronn A, Bucini G, Caylor KK, Coughenour MB, Diouf A, Ekaya W, Feral CJ, February EC, Frost PGH, Hiernaux P, Hrabar H, Metzger KL, Prins HHT, Ringrose S, Sea W, Tews J, Worden J, Zambatis N (2005) Determinants of woody covering African savannas. Nature 438:846–849CrossRefGoogle Scholar
  66. Sankaran M, Ratnam J, Hanan N (2008) Woody cover in African Savannas: the role of resources, fire and herbivory. Glob Ecol Biogeogr 17:236–245CrossRefGoogle Scholar
  67. Scheiter S, Higgins SI (2007) Partitioning of root and shoot competition and the stability of Savannas. Am Nat 170:587–601CrossRefGoogle Scholar
  68. Scholes RJ (2003) Convex relationships in ecosystems containing mixtures of trees and grass. Environ Ressour Econ 26:559–574CrossRefGoogle Scholar
  69. Scholes RJ, Archer SR (1997) tree-grass interactions in savannas. Annu Rev Ecol Syst 28:517–544CrossRefGoogle Scholar
  70. Scholes RJ, Walker BH (1993) An African savanna: synthesis of the Nylsvley study. Cambridge Studies in Applied Ecology and Resource Management. Cambridge University Press, Cambridge, UKGoogle Scholar
  71. Smit IPJ, Asner G, Govender N, Kennedy-Bowdoin T, Knapp D, Jacobson J (2010) Effects of fire on woody vegetation structure in African savanna. Ecol Appl 20(7):1865–1875CrossRefGoogle Scholar
  72. Smit GN, Rethman NFG (2000) The influence of tree thinning on the soil water in a semi-arid savanna of southern Africa. J Arid Environ 44:41–59CrossRefGoogle Scholar
  73. Sonntag Y (1997) Topologie et analyse fonctionnelle. Cours de Licence avec 240 exercices et 30 problèmes corrigés. Ellipses Collection Université, p 512Google Scholar
  74. Staver AC, Archibald S, Levin S (2011) Tree cover in sub-Saharan Africa: rainfall and fire constrain forest and Savanna as alternative stable states. Ecology 92(5):1063–1072CrossRefGoogle Scholar
  75. Staver AC, Bond WJ (2014) Is there a browse trap ? Dynamics of herbivore impacts on trees and grasses in an African savannas. J Ecol 102:595–602CrossRefGoogle Scholar
  76. Staver AC, Levin S (2012) Integrating theoretical climate and fire effects on savanna and forest systems. Am Nat 180(2):211–224CrossRefGoogle Scholar
  77. Synodinos AD, Tietjen B, Jeltsch F (2015) Facilitation in drylands: modeling a neglected driver of Savanna dynamics. Ecol Model 304:11–21CrossRefGoogle Scholar
  78. Tchuinté A, Tewa JJ, Couteron P, Bowong S, Dumont Y (2014) A generic modeling of fire impact in a tree-grass Savanna model. Biomath 3:1407191MathSciNetCrossRefzbMATHGoogle Scholar
  79. Tilman D (1994) Competition and biodiversity in spatially structured habitats. Ecology 75:2–16CrossRefGoogle Scholar
  80. Trollope WSW (1984) Fire in savannas. In: de Booysen PV, Tainton NM (eds) Ecological effects of fire in South African ecosystems. Ecological Studies series, vol 48. Spinger-Verlag, Berlin, pp 149–176Google Scholar
  81. Trollope WSW (1996) Behaviour, effects and use of fire in the savannas of southern Africa. In: Grice TC, Slatter SM (eds) Fire in the management of northern Australian pastoral lands. Proceedings of the Tropical Grassland Society of Australia, vol 8, pp 9–23Google Scholar
  82. Trollope WSW, and Trollope LA (1996) Fire in African savanna and other grazing ecosystems. Paper presented at the seminar on ’Forest fire and global change’ held in Shshenkoye in the Russian Federation, 4–10 AugustGoogle Scholar
  83. Tschinkel WR (2012) The life cycle and life span of Namibian fairy circles. PLoS One 7(6):e38056CrossRefGoogle Scholar
  84. Van de Vijver CA, Foley Olff H (1999) Changes in the woody component of an East African savanna during 25 years. J Trop Ecol 15:545–564CrossRefGoogle Scholar
  85. Van Langevelde F, Van de Vijver C, Kumar L, Van de Koppel J, de Ridder N, Van Andel J et al (2003) Effects of fire and herbivory on the stability of savanna ecosystems. Ecology 84(2):337–350CrossRefGoogle Scholar
  86. Van Wilgen BW, Govender N, Biggs HC, Ntsala D, Funda XN (2004) Response of savanna fire regimes to changing fire-management policies in a large African national park. Conserv Biol 18(6):1537–1540Google Scholar
  87. Wakeling JL, Staver AC, Bond WJ (2011) Simply the best: the transition of savanna saplings to trees. Oikos 120:1448–1451CrossRefGoogle Scholar
  88. Walker B, Ludwig D, Holling CS, Peterman RM (1981) Stability of semi-arid savanna grazing systems. J Ecol 69:473–498CrossRefGoogle Scholar
  89. Weltzin JF, Coughenour MB (1990) Savanna tree influence on understory vegetation and soil nutrients in northwestern Kenya. J Veg Sci 1:325–334CrossRefGoogle Scholar
  90. Yatat V, Dumont Y, Tewa JJ, Couteron P, Bowong S (2014) Mathematical analysis of a size structured tree-grass competition model for Savanna ecosystems. Biomath 3:1404212MathSciNetCrossRefGoogle Scholar

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
    Email author
  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

Personalised recommendations