Compartmental Models of Migratory Dynamics

J. Knisley; T. Schmickl; I. Karsai

Mathematical Modelling of Natural Phenomena (2011)

  • Volume: 6, Issue: 6, page 245-259
  • ISSN: 0973-5348

Abstract

top
Compartmentalization is a general principle in biological systems which is observable on all size scales, ranging from organelles inside of cells, cells in histology, and up to the level of groups, herds, swarms, meta-populations, and populations. Compartmental models are often used to model such phenomena, but such models can be both highly nonlinear and difficult to work with.Fortunately, there are many significant biological systems that are amenable to linear compartmental models which are often more mathematically accessible. Moreover, the biology and mathematics is often so intertwined in such models that one can be used to better understand the other. Indeed, as we demonstrate in this paper, linear compartmental models of migratory dynamics can be used as an exciting and interactive means of introducing sophisticated mathematics, and conversely, the associated mathematics can be used to demonstrate important biological properties not only of seasonal migrations but also of compartmental models in general.We have found this approach to be of great value in introducing derivatives, integrals, and the fundamental theorem of calculus. Additionally, these models are appropriate as applications in a differential equations course, and they can also be used to illustrate important ideas in probability and statistics, such as the Poisson distribution.

How to cite

top

Knisley, J., Schmickl, T., and Karsai, I.. "Compartmental Models of Migratory Dynamics." Mathematical Modelling of Natural Phenomena 6.6 (2011): 245-259. <http://eudml.org/doc/222293>.

@article{Knisley2011,
abstract = {Compartmentalization is a general principle in biological systems which is observable on all size scales, ranging from organelles inside of cells, cells in histology, and up to the level of groups, herds, swarms, meta-populations, and populations. Compartmental models are often used to model such phenomena, but such models can be both highly nonlinear and difficult to work with.Fortunately, there are many significant biological systems that are amenable to linear compartmental models which are often more mathematically accessible. Moreover, the biology and mathematics is often so intertwined in such models that one can be used to better understand the other. Indeed, as we demonstrate in this paper, linear compartmental models of migratory dynamics can be used as an exciting and interactive means of introducing sophisticated mathematics, and conversely, the associated mathematics can be used to demonstrate important biological properties not only of seasonal migrations but also of compartmental models in general.We have found this approach to be of great value in introducing derivatives, integrals, and the fundamental theorem of calculus. Additionally, these models are appropriate as applications in a differential equations course, and they can also be used to illustrate important ideas in probability and statistics, such as the Poisson distribution. },
author = {Knisley, J., Schmickl, T., Karsai, I.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {compartmental models; erlang; poisson; exponential; Erlang; Poisson},
language = {eng},
month = {10},
number = {6},
pages = {245-259},
publisher = {EDP Sciences},
title = {Compartmental Models of Migratory Dynamics},
url = {http://eudml.org/doc/222293},
volume = {6},
year = {2011},
}

TY - JOUR
AU - Knisley, J.
AU - Schmickl, T.
AU - Karsai, I.
TI - Compartmental Models of Migratory Dynamics
JO - Mathematical Modelling of Natural Phenomena
DA - 2011/10//
PB - EDP Sciences
VL - 6
IS - 6
SP - 245
EP - 259
AB - Compartmentalization is a general principle in biological systems which is observable on all size scales, ranging from organelles inside of cells, cells in histology, and up to the level of groups, herds, swarms, meta-populations, and populations. Compartmental models are often used to model such phenomena, but such models can be both highly nonlinear and difficult to work with.Fortunately, there are many significant biological systems that are amenable to linear compartmental models which are often more mathematically accessible. Moreover, the biology and mathematics is often so intertwined in such models that one can be used to better understand the other. Indeed, as we demonstrate in this paper, linear compartmental models of migratory dynamics can be used as an exciting and interactive means of introducing sophisticated mathematics, and conversely, the associated mathematics can be used to demonstrate important biological properties not only of seasonal migrations but also of compartmental models in general.We have found this approach to be of great value in introducing derivatives, integrals, and the fundamental theorem of calculus. Additionally, these models are appropriate as applications in a differential equations course, and they can also be used to illustrate important ideas in probability and statistics, such as the Poisson distribution.
LA - eng
KW - compartmental models; erlang; poisson; exponential; Erlang; Poisson
UR - http://eudml.org/doc/222293
ER -

References

top
  1. M. Allarakhia, A. Wensley. Systems biology: melting the boundaries in drug discovery research. Technology Management: A Unifying Discipline for Melting the Boundaries, (2005), 262–274.  
  2. L. Allen. An Introduction to mathematical biology, Pearson, New York, 2007.  
  3. S. Bauer, Z. Barta, B. Ens, G. Hays, J. McNamara, M. Klassen. Animal migration: linking models and data beyond taxonomic limits. Biol. Lett., 5 (2009), No. 4, 433–435.  
  4. R. Baxter. Environmental effects of dams and impoundments. Ann. Rev. Ecol. Syst., 8 (1977), 255–93.  
  5. W. Bialek, D. Botstein. Introductory science and mathematics education for the 21th century biologists. Science, 303 (2004), 788–790.  
  6. J. Bower. Looking for Newton: realistic modeling in modern biology. Brains, Minds and Media, 1 (2005), bmm217 (urn:nbn:de:0009-3-2177).  
  7. V. Buonaccorsi, A. Skibiel. A striking demonstration of the Poisson distribution. Teach. Stat, 27 (2005), 8-10.  
  8. C. Cobelli. Modeling and identification of endocrine-metabolic systems. Theoretical aspects and their importance in practice. Math. Biosci., 72 (1984), 263–289.  
  9. J. Cohen. Mathematics Is Biology’s Next Microscope, Only Better; Biology Is Mathematics’ Next Physics, Only Better. PLoS Biol., 2 (2004), e439. Published online 2004 December 14. doi: . URI10.1371/journal.pbio.0020439
  10. J. Crow and M. Kimura. An Introduction to Population Genetics Theory. Harper & Row, New York, 1970.  
  11. M. Evans, N. Hastings, B. Peacock. Erlang distribution. Ch. 12 in Statistical Distributions, 3rd ed., Wiley, New York, 2000, 71–73.  
  12. A. Ford. Modeling the environment. Island Press, 2010.  
  13. W. Granta, J. Matisb, and T. Millerb. A Stochastic Compartmental Model for Migration of Marine Shrimp. Ecological Modeling, 54 (1991), 1–15.  
  14. L. Gross. Quantitative training for life science students. BioScience, 44 (1994), 59.  
  15. P. Higgs. Frequency distributions in population genetics parallel those in statistical physics. Physical Review E, (1995), No. 51, 95–101.  
  16. W. Hwang, Y. Cho, A. Zhang, M. Ramanathan. A novel functional module detection algorithm for protein-protein interaction networks. Algorithms for Molecular Biology (2006), No. 2, 24, doi:.  URI10.1186/1748-7188-1-24
  17. J. Jacquez. Compartmental Analysis in Biology and Medicine, 3rd ed., Biomedware, Ann Arbor, MI, 1996.  
  18. I. Karsai, G. Kampis. The crossroad between biology and mathematics: Scientific method as the basics of scientific literacy. BioScience, 60 (2010), 632–638.  
  19. J. Knisley. Netlogo Migration Simulations, , (2011).  URIhttp://math.etsu.edu/symbiosis/migrations
  20. J. Knisley, I. Karsai, A. Godbole, M. Helfgott, K. Joplin, E. Seier, D. Moore, H. Miller. Storytelling in the Symbiosis Project. To appear in Undergraduate Mathematics for the Life Sciences: Processes, Models, Assessment, and Directions, MAA Lecture Notes, 2010.  
  21. J. Knisley. A 4-Stage Model of Mathematical Learning. The Mathematics Educator, 12 (2002), No. 1, 11–16.  
  22. D. Lauffenburger. Receptors. Oxford University Press, Oxford, 1993.  
  23. M. Malice, C. Lefevre. On Linear Stochastic Compartmental Models in Discrete Time. Bulletin of Mathematical Biology, 47 (1985), No. 2, 287–293.  
  24. D. Moore, M. Helfgott, A. Godbole, K. Joplin, I. Karsai, J. Knisley, H. Miller, E. Seier. Creating Quantitative Biologists: The Immediate Future of SYMBIOSIS. To appear in Undergraduate Mathematics for the Life Sciences: Processes, Models, Assessment, and Directions, MAA Lecture Notes, 2010.  
  25. J. Murray. Mathematical Biology. Springer, New York, 1989.  
  26. W. Reed, B. Hughes. Theoretical size distribution of fossil taxa: analysis of a null model. Theoretical Biology and Medical Modeling, 4 (2007), 12, doi: URI10.1186/1742-4682-4-12
  27. L. Steen, ed. Math & Bio 2010: Linking Undergraduate Disciplines. Mathematical Association of America, Washington, DC, 2005.  
  28. D. Usher, T. Driscoll, P. Dhurjati, J. Pelesko, L. Rossi, G. Schleiniger, K. Pusecker, H. White. A transformative model for undergraduate quantitative biology education. CBE Life Sci Educ., 9 (2010), No. 3, 181–188.  
  29. D. Welch, E. Rechisky, M. Melnychuk, A. Porter, C. Walters. Survival of Migrating Salmon Smolts in Large Rivers With and Without Dams. PLoS Biol, 6 (2008), No. 10, e265. doi:.  URI10.1371/journal.pbio.0060265
  30. U. Wilensky. NetLogo. . Center for Connected Learning and Computer-Based Modeling, Northwestern University. Evanston, IL, (1999).  URIhttp://ccl.northwestern.edu/netlogo/

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.