Mathematical modeling of forest ecosystems by a reaction-diffusion-advection system: impacts of climate change and deforestation. (English) Zbl 1480.92219

Summary: We present an innovative mathematical model for studying the dynamics of forest ecosystems. Our model is determined by an age-structured reaction-diffusion-advection system in which the roles of the water resource and of the atmospheric activity are considered. The model is abstract but constructed in such a manner that it can be applied to real-world forest areas; thus it allows to establish an infinite number of scenarios for testing the robustness and resilience of forest ecosystems to anthropic actions or to climate change. We establish the well-posedness of the reaction-diffusion-advection model by using the method of characteristics and by reducing the initial system to a reaction-diffusion problem. The existence and stability of stationary homogeneous and stationary heterogeneous solutions are investigated, so as to prove that the model is able to reproduce relevant equilibrium states of the forest ecosystem. We show that the model fits with the principle of almost uniform precipitation over forested areas and of exponential decrease of precipitation over deforested areas. Furthermore, we present a selection of numerical simulations for an abstract forest ecosystem, in order to analyze the stability of the steady states, to investigate the impact of anthropic perturbations such as deforestation and to explore the effects of climate change on the dynamics of the forest ecosystem.


92D40 Ecology
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs


FreeFem++; SORTIE
Full Text: DOI


[1] Anderegg, WR; Trugman, AT; Badgley, G.; Konings, AG; Shaw, J., Divergent forest sensitivity to repeated extreme droughts, Nat Clim Change, 10, 1-5 (2020)
[2] Antonovsky, MY; Fleming, R.; Kuznetsov, YA; Clark, W., Forest-pest interaction dynamics: the simplest mathematical models, Theor Popul Biol, 37, 2, 343-367 (1990) · Zbl 0699.92023
[3] Botkin DB et al (1993) Forest dynamics: an ecological model. Oxford University Press on Demand
[4] Cantin, G.; Verdière, N., Networks of forest ecosystems: mathematical modeling of their biotic pump mechanism and resilience to certain patch deforestation, Ecol Complex, 43, 100850 (2020)
[5] Cantrell, RS; Cosner, C., Spatial ecology via reaction-diffusion equations (2004), Hoboken: Wiley, Hoboken · Zbl 1059.92051
[6] Cook-Patton, SC; Leavitt, SM; Gibbs, D.; Harris, NL; Lister, K.; Anderson-Teixeira, KJ; Briggs, RD; Chazdon, RL; Crowther, TW; Ellis, PW, Mapping carbon accumulation potential from global natural forest regrowth, Nature, 585, 7826, 545-550 (2020)
[7] Cosandey C (1992) Influence de la forêt sur le cycle de l’eau: conséquences d’une coupe forestière sur le bilan d’écoulement annuel. Hydrologie continentale 7(1):13-22
[8] Creed, IF; Weber, M.; Accatino, F.; Kreutzweiser, DP, Managing forests for water in the anthropocene-the best kept secret services of forest ecosystems, Forests, 7, 3, 60 (2016)
[9] Dubreuil V, Arvor D, Funatsu BM, Nédélec V, Mello-Théry NAd (2020) Les changements climatiques en Amazonie, une approche multiscalaire. In: Mercier D (ed) Les impacts spatiaux du changement climatique. ISTE, pp 247-270
[10] Fernandes, K.; Giannini, A.; Verchot, L.; Baethgen, W.; Pinedo-Vasquez, M., Decadal covariability of Atlantic SSTs and western Amazon dry-season hydroclimate in observations and CMIP5 simulations, Geophys Res Lett, 42, 16, 6793-6801 (2015)
[11] Filippov, AF, Differential equations with discontinuous righthand sides: control systems (2013), Berlin: Springer, Berlin
[12] Flannigan, MD; Stocks, BJ; Wotton, BM, Climate change and forest fires, Sci Total Environ, 262, 3, 221-229 (2000)
[13] Gillett, N.; Weaver, A.; Zwiers, F.; Flannigan, M., Detecting the effect of climate change on Canadian forest fires, Geophys Res Lett, 31, 18, L18211 (2004)
[14] Hansen AJ, Burns P, Ervin J, Goetz SJ, Hansen M, Venter O, Watson JE, Jantz PA, Virnig AL, Barnett K et al (2020) A policy-driven framework for conserving the best of earth’s remaining moist tropical forests. Nat Ecol Evol 41:1-8
[15] Hecht F, Pironneau O, Le Hyaric A, Ohtsuka K (2005) Freefem++ manual
[16] Henry, D., Geometric theory of semilinear parabolic equations (2006), Berlin: Springer, Berlin
[17] Keenan, RJ, Climate change impacts and adaptation in forest management: a review, Ann For Sci, 72, 2, 145-167 (2015)
[18] Kohyama, T., Size-structured multi-species model of rain forest trees, Funct Ecol, 6, 206-212 (1992)
[19] Kolobov, A.; Frisman, EY, Individual-based model of spatio-temporal dynamics of mixed forest stands, Ecol Complex, 27, 29-39 (2016)
[20] Kuznetsov, YA; Antonovsky, MY; Biktashev, V.; Aponina, E., A cross-diffusion model of forest boundary dynamics, J Math Biol, 32, 3, 219-232 (1994) · Zbl 0790.92028
[21] Le Huy, C.; Tsujikawa, T.; Yagi, A., Asymptotic behavior of solutions for forest kinematic model, Funkcialaj Ekvacioj, 49, 3, 427-449 (2006) · Zbl 1157.35317
[22] Le Huy, C.; Tsujikawa, T.; Yagi, A., Stationary solutions to forest kinematic model, Glasgow Math J, 51, 1, 1-17 (2009) · Zbl 1152.35366
[23] Magal, P.; Zhang, Z., Competition for light in forest population dynamics: from computer simulator to mathematical model, J Theor Biol, 419, 290-304 (2017) · Zbl 1370.92140
[24] Nobre CA, Obregón GO, Marengo JA, Fu R, Poveda G (2009) Characteristics of Amazonian climate: main features. In: Amazonia and global change. American Geophysical Union, pp 149-162 (AGU)
[25] North, M.; Stephens, S.; Collins, B.; Agee, J.; Aplet, G.; Franklin, J.; Fule, PZ, Reform forest fire management, Science, 349, 6254, 1280-1281 (2015)
[26] Noss, RF, Beyond Kyoto: forest management in a time of rapid climate change, Conserv Biol, 15, 3, 578-590 (2001)
[27] Oyama, MD; Nobre, CA, A new climate-vegetation equilibrium state for Tropical South America, Geophys Res Lett, 30, 23, 2199 (2003)
[28] Running, SW; Nemani, RR; Peterson, DL; Band, LE; Potts, DF; Pierce, LL; Spanner, MA, Mapping regional forest evapotranspiration and photosynthesis by coupling satellite data with ecosystem simulation, Ecology, 70, 4, 1090-1101 (1989)
[29] Schertzer, E.; Staver, A.; Levin, SA, Implications of the spatial dynamics of fire spread for the bistability of savanna and forest, J Math Biol, 70, 1-2, 329-341 (2015) · Zbl 1309.92081
[30] Schwetlick, HR, Limit sets for multidimensional nonlinear transport equations, J Differ Equ, 179, 1, 356-368 (2002) · Zbl 1003.35078
[31] Shirai, T.; Chuan, L.; Yagi, A., Dynamical system for forest kinematic model under Dirichlet conditions, Scientiae Mathematicae japonicae, 66, 2, 275-288 (2007) · Zbl 1152.35396
[32] Shukla, J.; Dubey, B., Modelling the depletion and conservation of forestry resources: effects of population and pollution, J Math Biol, 36, 1, 71-94 (1997) · Zbl 0893.92035
[33] Smith, HL; Thieme, HR, Dynamical systems and population persistence (2011), Providence: American Mathematical Society, Providence · Zbl 1214.37002
[34] Spittlehouse DL, Stewart RB (2004) Adaptation to climate change in forest management. J Ecosyst Manag 4(1)
[35] Strang, G., On the construction and comparison of difference schemes, SIAM J Numer Anal, 5, 3, 506-517 (1968) · Zbl 0184.38503
[36] Veldkamp, E.; Schmidt, M.; Powers, JS; Corre, MD, Deforestation and reforestation impacts on soils in the tropics, Nat Rev Earth Environ, 1, 1-16 (2020)
[37] Verzelen, N.; Picard, N.; Gourlet-Fleury, S., Approximating spatial interactions in a model of forest dynamics as a means of understanding spatial patterns, Ecol Complex, 3, 3, 209-218 (2006)
[38] Webb, T., Is vegetation in equilibrium with climate? How to interpret late-Quaternary pollen data, Vegetatio, 67, 2, 75-91 (1986)
[39] Wilson, KB; Hanson, PJ; Mulholland, PJ; Baldocchi, DD; Wullschleger, SD, A comparison of methods for determining forest evapotranspiration and its components: sap-flow, soil water budget, eddy covariance and catchment water balance, Agric For Meteorol, 106, 2, 153-168 (2001)
[40] Yagi, A., Abstract parabolic evolution equations and their applications (2009), Berlin: Springer, Berlin · Zbl 1190.35004
[41] Yatat, V.; Couteron, P.; Tewa, JJ; Bowong, S.; Dumont, Y., An impulsive modelling framework of fire occurrence in a size-structured model of tree-grass interactions for savanna ecosystems, J Math Biol, 74, 6, 1425-1482 (2017) · Zbl 1366.34069
[42] Yousefpour, R.; You, B.; Hanewinkel, M., Simulation of extreme storm effects on regional forest soil carbon stock, Ecol Model, 399, 39-53 (2019)
[43] Zemp, DC; Schleussner, C-F; Barbosa, HM; Hirota, M.; Montade, V.; Sampaio, G.; Staal, A.; Wang-Erlandsson, L.; Rammig, A., Self-amplified Amazon forest loss due to vegetation-atmosphere feedbacks, Nat Commun, 8, 1, 1-10 (2017)
[44] Zheng, Z.; Huang, W.; Li, S.; Zeng, Y., Forest fire spread simulating model using cellular automaton with extreme learning machine, Ecol Model, 348, 33-43 (2017)
[45] Zhou, G.; Meng, C.; Jiang, P.; Xu, Q., Review of carbon fixation in bamboo forests in China, Bot Rev, 77, 3, 262 (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.