Figure 1.
Time evolution of $R$ in three different cases: (a) $g = 1$; (b) $g = 0.2$ between $t = 3$ and $t = 10$ and $g = 1$ elsewhere, (c) $g = 1$ if $t\le 3$ and $g = 0.2$ elsewhere
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A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay
An age group model for the study of a population of trees
| 1. | LMR, UMR 9008, U.F.R. Sciences Exactes et Naturelles, BP 1039, 51687 Reims Cedex 02, France |
| 2. | HABITER, EA 2076, U.F.R. Lettres et Sciences Humaines, BP 30, 51571 Reims Cedex, France |
In this paper, we derive a simple model for the description of an ecological system including several subgroups with distinct ages, in order to analyze the influence of various phenomena on temporal evolution of the considered species. Our aim is to address the question of resilience of the global system, defined as its ability to stabilize itself to equilibrium, when being perturbed by exterior fluctuations. It is shown that a under a critical condition involving growth rate and mortality rate of each subgroup, extinction of all species may occur.
Mathematics Subject Classification:
Primary: 34D20; Secondary: 65L05, 92D25.
Citation:
Laurent Di Menza, Virginie Joanne-Fabre.
An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020464
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References:
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O. Y. Achiepo, Les bases fondamentales de la résilométrie, une science de modélisation de la souffrance & Les dimensions de la mesure de la résilience, Cafés de Résiliences (2015). Google Scholar |
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W. N. Adger,
Vulnerability, Global Environmental Change, 16 (2006), 268-281.
doi: 10.1016/j.gloenvcha.2006.02.006. |
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P. Auger, C. Lett and J.-C. Poggiale, Modélisation Mathématique en Ecologie, Sciences Sup, SMAI/Dunod, 2015. Google Scholar |
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B. E. Beisner, D. T. Haydon and K. Cuddington, Alternative stable states in ecology, Front. Ecol. Environ., 1, (2003), 376–382.http://dx.doi.org/10.1890/1540-9295(2003)001[0376:ASSIE]2.0.CO;2 Google Scholar |
| [5] |
F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New-York, 2001.
doi: 10.1007/978-1-4757-3516-1. |
| [6] |
P. Buchheit, P. d'Aquino and O. Ducourtieux, Cadres théoriques mobilisant les concepts de résilience et de vulnérabilité, VertigO, 16 (2016).
doi: 10.4000/vertigo.17131. |
| [7] |
G. Cantin and N. Verdière, Mathematical modeling of complex forest ecosystems: Impacts of deforestation, (2020). [hal-02496187]. Google Scholar |
| [8] |
S. Carpenter, B. Walker, J. M. Anderies and N. Abel,
From metaphor to measurement: Resilience of what to what?, Ecosystems, 4 (2001), 765-781.
doi: 10.1007/s10021-001-0045-9. |
| [9] |
B. Courbaud, Intérêt des modèles pour l'ingénierie écologique: Exemples à partir du modèle de dynamique des peuplements forestiers SAMSARA, in Ingénieries - EAT, Ingénierie écologique, Special issue (2004), 49–56. Google Scholar |
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V. Fabre, Réponse Démographique des Néandertaliens Face Aux Pressions Environnementales du Stade Isotopique 3 : Approche Par Modélisation Écologique, Ph. D Thesis, Marseille University, France, 2011. Google Scholar |
| [11] |
V. Fabre and J.-C. Poggiale, Modeling the resilience of past populations: Myth or real 6 perspective? Application to Neanderthals demise, personal publication (2016). Google Scholar |
| [12] |
C. Folke,
Resilience: The emergence of a perspective for social-ecological systems analysis, Global Environmental Change, 16 (2006), 253-267.
doi: 10.1016/j.gloenvcha.2006.04.002. |
| [13] |
C. Folke,
Social-ecological systems and adaptive governance of the commons, Ecological Research, 22 (2007), 14-15.
doi: 10.1007/s11284-006-0074-0. |
| [14] |
C. Folke, S. R. Carpenter, B. Walker, M. Scheffer, T. Chapin and J. Rockström, Resilience thinking: Integrating resilience, adaptability and transformability, Ecology and Society, 15 (2010), 20.
doi: 10.5751/ES-03610-150420. |
| [15] |
E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equations: Non-Stiff Problems, Springer Series in Computational Mathematics, Springer-Verlag, 2nd Ed., 1993. |
| [16] |
C. S. Holling,
Resilience and stability of ecological systems, Annual Rev. of Ecology and Systematics, 4 (1973), 1-23.
doi: 10.1146/annurev.es.04.110173.000245. |
| [17] |
C. S. Holling, The resilience of terrestrial ecosystems: Local surprise and global change, in Sustainable Development of the Biosphere, Clark W. C., Munn R. E. Eds, Cambridge University Press, London (1986), 292–317. Google Scholar |
| [18] |
C. S. Holling,
Understanding the complexity of economic, ecological, and social systems, Ecosystems, 4 (2001), 390-405.
doi: 10.1007/s10021-001-0101-5. |
| [19] |
A. Hurwitz, Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen T eilen besitzt, Math. Annalen, 46, (1895), 273–284.
doi: 10.1007/BF01446812. |
| [20] |
S. A. Levin and J. Lubchenco, Resilience, Robustness, and Marine Ecosystem-Based Management, BioScience, 58, (2008), 27–32.
doi: 10.1641/B580107. |
| [21] |
J. Liu, T. Dietz, S. R. Carpenter, M. Alberti, C. Folke, E. Moran, A. N. Pell, P. Deadman, T. Kratz, J. Lubchenco, E. Ostrom, Z. Ouyang, W. Provencher, C. L. Redman, S. H. Schneider and W. W. Taylor,
Complexity of coupled human and natural systems, Science, 317 (2007), 1513-1516.
doi: 10.1126/science.1144004. |
| [22] |
S. Martin, La résilience Dans Les Modèles de Systèmes Écologiques et Sociaux, Ph. D Thesis, ENS Cachan, France, 2005. Google Scholar |
| [23] |
K. Meyer,
A mathematical review of resiliene in ecology, Natural Resource Modeling, 29 (2016), 339-352.
doi: 10.1111/nrm.12097. |
| [24] |
F. Miller, H. Osbahr, E. Boyd, F. Thomalla, S. Bharwani, G. Ziervogel, B. Walker, J. Birkmann, S. Van Der Leeuw, J. Rockström, J. Hinkel, T. Downing, C. Folke and D. Nelson,
Resilience and Vulnerability: Complementary or Conflicting Concepts?, Ecology and Society, 15 (2010), 1-25.
doi: 10.5751/ES-03378-150311. |
| [25] |
N. Oreskes, The role of quantitative models in science, in Models in Ecosystem Science, Canham, Cole and Lauenroth Eds, W.-K, Princeton University Press, Princeton, (2003), 13–31. Google Scholar |
| [26] |
E. Ostrom,
A general framework for analyzing sustainability of social-ecological systems, Science, 325 (2009), 419-422.
doi: 10.1126/science.1172133. |
| [27] |
S. L. Pimm,
The complexity and stability of ecosystems, Nature, 307 (1984), 321-326.
doi: 10.1038/307321a0. |
| [28] |
C. L. Redman, J. M. Grove and L. H. Kuby,
Integrating social science into the Long-Term Ecological Research (LTER) network: Social dimensions of ecological change and ecological dimensions of social change, Ecosystems, 7 (2004), 161-171.
doi: 10.1007/s10021-003-0215-z. |
| [29] |
C. Rouge, J. D. Mathias and G. Deffuant,
Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication, Ecol. Indic., 29 (2013), 420-433.
doi: 10.1016/j.ecolind.2012.12.032. |
| [30] |
E. Specht, T. Redemann and N. Lorenz, Simplified mathematical model for calculating global warming through anthropogenic CO$_2$, International Journal of Thermal Sciences, 102 (2016), 1-8. Google Scholar |
| [31] |
B. L. Turner, R. E. Kasperson, P. A. Matson, J. J. McCarthy, R. W. Corell, L. Christensen, N. Eckley, J. X. Kasperson, A. Luers, M. L. Martello, C. Polsky, A. Pulsipher and A. Schiller,
A framework for vulnerability analysis in sustainability science, PNAS, 100 (2003), 8074-8079.
doi: 10.1073/pnas.1231335100. |
| [32] |
P.-F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121. Google Scholar |
| [33] |
V. Volterra,
Variations and Fluctuations of the Number of Individuals
in Animal Species Living Together, ICES Journal of Marine Science, 3 (1928), 3-51.
doi: 10.1093/icesjms/3.1.3. |
| [34] |
B. Walker, C. S. Holling, S. R. Carpenter and A. Kinzig, Resilience, adaptability and transformability in social-ecological systems, Ecol. Soc., 9 (2004), 5.
doi: 10.5751/ES-00650-090205. |
show all references
References:
| [1] |
O. Y. Achiepo, Les bases fondamentales de la résilométrie, une science de modélisation de la souffrance & Les dimensions de la mesure de la résilience, Cafés de Résiliences (2015). Google Scholar |
| [2] |
W. N. Adger,
Vulnerability, Global Environmental Change, 16 (2006), 268-281.
doi: 10.1016/j.gloenvcha.2006.02.006. |
| [3] |
P. Auger, C. Lett and J.-C. Poggiale, Modélisation Mathématique en Ecologie, Sciences Sup, SMAI/Dunod, 2015. Google Scholar |
| [4] |
B. E. Beisner, D. T. Haydon and K. Cuddington, Alternative stable states in ecology, Front. Ecol. Environ., 1, (2003), 376–382.http://dx.doi.org/10.1890/1540-9295(2003)001[0376:ASSIE]2.0.CO;2 Google Scholar |
| [5] |
F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New-York, 2001.
doi: 10.1007/978-1-4757-3516-1. |
| [6] |
P. Buchheit, P. d'Aquino and O. Ducourtieux, Cadres théoriques mobilisant les concepts de résilience et de vulnérabilité, VertigO, 16 (2016).
doi: 10.4000/vertigo.17131. |
| [7] |
G. Cantin and N. Verdière, Mathematical modeling of complex forest ecosystems: Impacts of deforestation, (2020). [hal-02496187]. Google Scholar |
| [8] |
S. Carpenter, B. Walker, J. M. Anderies and N. Abel,
From metaphor to measurement: Resilience of what to what?, Ecosystems, 4 (2001), 765-781.
doi: 10.1007/s10021-001-0045-9. |
| [9] |
B. Courbaud, Intérêt des modèles pour l'ingénierie écologique: Exemples à partir du modèle de dynamique des peuplements forestiers SAMSARA, in Ingénieries - EAT, Ingénierie écologique, Special issue (2004), 49–56. Google Scholar |
| [10] |
V. Fabre, Réponse Démographique des Néandertaliens Face Aux Pressions Environnementales du Stade Isotopique 3 : Approche Par Modélisation Écologique, Ph. D Thesis, Marseille University, France, 2011. Google Scholar |
| [11] |
V. Fabre and J.-C. Poggiale, Modeling the resilience of past populations: Myth or real 6 perspective? Application to Neanderthals demise, personal publication (2016). Google Scholar |
| [12] |
C. Folke,
Resilience: The emergence of a perspective for social-ecological systems analysis, Global Environmental Change, 16 (2006), 253-267.
doi: 10.1016/j.gloenvcha.2006.04.002. |
| [13] |
C. Folke,
Social-ecological systems and adaptive governance of the commons, Ecological Research, 22 (2007), 14-15.
doi: 10.1007/s11284-006-0074-0. |
| [14] |
C. Folke, S. R. Carpenter, B. Walker, M. Scheffer, T. Chapin and J. Rockström, Resilience thinking: Integrating resilience, adaptability and transformability, Ecology and Society, 15 (2010), 20.
doi: 10.5751/ES-03610-150420. |
| [15] |
E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equations: Non-Stiff Problems, Springer Series in Computational Mathematics, Springer-Verlag, 2nd Ed., 1993. |
| [16] |
C. S. Holling,
Resilience and stability of ecological systems, Annual Rev. of Ecology and Systematics, 4 (1973), 1-23.
doi: 10.1146/annurev.es.04.110173.000245. |
| [17] |
C. S. Holling, The resilience of terrestrial ecosystems: Local surprise and global change, in Sustainable Development of the Biosphere, Clark W. C., Munn R. E. Eds, Cambridge University Press, London (1986), 292–317. Google Scholar |
| [18] |
C. S. Holling,
Understanding the complexity of economic, ecological, and social systems, Ecosystems, 4 (2001), 390-405.
doi: 10.1007/s10021-001-0101-5. |
| [19] |
A. Hurwitz, Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen T eilen besitzt, Math. Annalen, 46, (1895), 273–284.
doi: 10.1007/BF01446812. |
| [20] |
S. A. Levin and J. Lubchenco, Resilience, Robustness, and Marine Ecosystem-Based Management, BioScience, 58, (2008), 27–32.
doi: 10.1641/B580107. |
| [21] |
J. Liu, T. Dietz, S. R. Carpenter, M. Alberti, C. Folke, E. Moran, A. N. Pell, P. Deadman, T. Kratz, J. Lubchenco, E. Ostrom, Z. Ouyang, W. Provencher, C. L. Redman, S. H. Schneider and W. W. Taylor,
Complexity of coupled human and natural systems, Science, 317 (2007), 1513-1516.
doi: 10.1126/science.1144004. |
| [22] |
S. Martin, La résilience Dans Les Modèles de Systèmes Écologiques et Sociaux, Ph. D Thesis, ENS Cachan, France, 2005. Google Scholar |
| [23] |
K. Meyer,
A mathematical review of resiliene in ecology, Natural Resource Modeling, 29 (2016), 339-352.
doi: 10.1111/nrm.12097. |
| [24] |
F. Miller, H. Osbahr, E. Boyd, F. Thomalla, S. Bharwani, G. Ziervogel, B. Walker, J. Birkmann, S. Van Der Leeuw, J. Rockström, J. Hinkel, T. Downing, C. Folke and D. Nelson,
Resilience and Vulnerability: Complementary or Conflicting Concepts?, Ecology and Society, 15 (2010), 1-25.
doi: 10.5751/ES-03378-150311. |
| [25] |
N. Oreskes, The role of quantitative models in science, in Models in Ecosystem Science, Canham, Cole and Lauenroth Eds, W.-K, Princeton University Press, Princeton, (2003), 13–31. Google Scholar |
| [26] |
E. Ostrom,
A general framework for analyzing sustainability of social-ecological systems, Science, 325 (2009), 419-422.
doi: 10.1126/science.1172133. |
| [27] |
S. L. Pimm,
The complexity and stability of ecosystems, Nature, 307 (1984), 321-326.
doi: 10.1038/307321a0. |
| [28] |
C. L. Redman, J. M. Grove and L. H. Kuby,
Integrating social science into the Long-Term Ecological Research (LTER) network: Social dimensions of ecological change and ecological dimensions of social change, Ecosystems, 7 (2004), 161-171.
doi: 10.1007/s10021-003-0215-z. |
| [29] |
C. Rouge, J. D. Mathias and G. Deffuant,
Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication, Ecol. Indic., 29 (2013), 420-433.
doi: 10.1016/j.ecolind.2012.12.032. |
| [30] |
E. Specht, T. Redemann and N. Lorenz, Simplified mathematical model for calculating global warming through anthropogenic CO$_2$, International Journal of Thermal Sciences, 102 (2016), 1-8. Google Scholar |
| [31] |
B. L. Turner, R. E. Kasperson, P. A. Matson, J. J. McCarthy, R. W. Corell, L. Christensen, N. Eckley, J. X. Kasperson, A. Luers, M. L. Martello, C. Polsky, A. Pulsipher and A. Schiller,
A framework for vulnerability analysis in sustainability science, PNAS, 100 (2003), 8074-8079.
doi: 10.1073/pnas.1231335100. |
| [32] |
P.-F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121. Google Scholar |
| [33] |
V. Volterra,
Variations and Fluctuations of the Number of Individuals
in Animal Species Living Together, ICES Journal of Marine Science, 3 (1928), 3-51.
doi: 10.1093/icesjms/3.1.3. |
| [34] |
B. Walker, C. S. Holling, S. R. Carpenter and A. Kinzig, Resilience, adaptability and transformability in social-ecological systems, Ecol. Soc., 9 (2004), 5.
doi: 10.5751/ES-00650-090205. |
Figure 2.
Time evolution of $R$ in two different cases: (a) $m = 1$; (b) $m = 1.2$ between $t = 20$ and $t = 40$ and $m = 1$ elsewhere. The asymptotic state $R_{*, 2}$ is not perturbed by mortality fluctuation
Figure 3.
Time evolution of $R$ in two different cases: (a) $m = 1.2$ for $t\ge 20$; (b) $m = 2$ for $t\ge 20$. In the first case, the asymptotic state $R_{*, 2}$ is perturbed, in the second case, extinction occurs at large times
Figure 4.
Time evolution of $(Y, I, O)$ for the two stable cases $g_Y = 3$, $g_I = 1$, $g_O = 0.5$, $m_Y = 1$, $m_I = 0.8$, $m_O = 0.2$ (left) and $g_Y = 1$, $g_I = 1$, $g_O = 0.6$, $m_Y = 1.5$, $m_I = 1$, $m_O = 0.2$ (right) ($Y$: black, $I$: blue, $O$: red)
Figure 5.
Time evolution of $(Y, I, O)$ when perturbing mortality of intermediate trees : $m_{I, 2} = 1.4$ (left) and $m_{I, 2} = 2$ (right). In the first case, the system drives itself into another stable nontrivial equilibrium; in the second case, the total population extincts ($Y$: black, $I$: blue, $O$: red)
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