American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data
  • DCDS-S Home
  • This Issue
  • Next Article
    Numerical simulations of parity–time symmetric nonlinear Schrödinger equations in critical case
August  2021, 14(8): 2823-2835. doi: 10.3934/dcdss.2020464

An age group model for the study of a population of trees

Laurent Di Menza 1,, and  Virginie Joanne-Fabre 2,

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

 

Received  June 2020 Revised  October 2020 Published  August 2021 Early access  November 2020

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.

Keywords: Ecology, dynamical systems, stability analysis.
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, 2021, 14 (8) : 2823-2835. doi: 10.3934/dcdss.2020464
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. A. Levin and J. Lubchenco, Resilience, Robustness, and Marine Ecosystem-Based Management, BioScience, 58, (2008), 27–32. doi: 10.1641/B580107.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. L. Pimm, The complexity and stability of ecosystems, Nature, 307 (1984), 321-326.  doi: 10.1038/307321a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. A. Levin and J. Lubchenco, Resilience, Robustness, and Marine Ecosystem-Based Management, BioScience, 58, (2008), 27–32. doi: 10.1641/B580107.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. L. Pimm, The complexity and stability of ecosystems, Nature, 307 (1984), 321-326.  doi: 10.1038/307321a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317
[2]

Boris Paneah. Noncommutative dynamical systems with two generators and their applications in analysis. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1411-1422. doi: 10.3934/dcds.2003.9.1411
[3]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129
[4]

S.Durga Bhavani, K. Viswanath. A general approach to stability and sensitivity in dynamical systems. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 131-140. doi: 10.3934/dcds.1998.4.131
[5]

Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533
[6]

Kamaldeen Okuneye, Ahmed Abdelrazec, Abba B. Gumel. Mathematical analysis of a weather-driven model for the population ecology of mosquitoes. Mathematical Biosciences & Engineering, 2018, 15 (1) : 57-93. doi: 10.3934/mbe.2018003
[7]

Alexander Pimenov, Dmitrii I. Rachinskii. Linear stability analysis of systems with Preisach memory. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 997-1018. doi: 10.3934/dcdsb.2009.11.997
[8]

Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487
[9]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
[10]

J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467
[11]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115
[12]

Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571
[13]

Daniel Franco, Juan Perán, Juan Segura. Stability for one-dimensional discrete dynamical systems revisited. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 635-650. doi: 10.3934/dcdsb.2019258
[14]

Giovanni Russo, Fabian Wirth. Matrix measures, stability and contraction theory for dynamical systems on time scales. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021188
[15]

Ken Shirakawa. Stability analysis for phase field systems associated with crystalline-type energies. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 483-504. doi: 10.3934/dcdss.2011.4.483
[16]

Sibel Senan, Eylem Yucel, Zeynep Orman, Ruya Samli, Sabri Arik. A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1415-1428. doi: 10.3934/dcdss.2020358
[17]

Ken Shirakawa. Asymptotic stability for dynamical systems associated with the one-dimensional Frémond model of shape memory alloys. Conference Publications, 2003, 2003 (Special) : 798-808. doi: 10.3934/proc.2003.2003.798
[18]

Radosław Czaja. Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021276
[19]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449
[20]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (144)
  • HTML views (303)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy