MATH 5740/MATH 6870 001 MATH MODELING
MWF 09:40 AM-10:30 AM JWB 208
Three credit hours
Instructor: Professor Andrej Cherkaev Department of Mathematics Office: JWB 225 Email: cherk@math.utah.edu Tel : +1 801 - 581 6822 |
Note 1. Population dynamics models
Note 2. Leslie Population dynamics model http://www.math.utah.edu/~cherk/teach/12mathmodel/leslie-model.pdf
Research the models of population dynamics (see Note 1).
Simulate population dynamics (you may use the codes in
note 1) in different models as differential and difference
equations. Simulate the diffusive population growth
using difference scheme. Assume periodic boundary
conditions,
Due Wednesday, Jan. 16.
2.1. Find the data: age-dependent rate of birth and
surviving in a specific community (city, state, country)
Determine the asymptotic age distribution, based on the
Leslie matrix analysis. Run the model, suggest an
adjustment to account for limiting habitat capacity.
Suggest a modele and simulate the age-dependent migration
between two communities.
2.2. Suggest a predator-prey model for two predator species and one prey species. Find stable points. Are they stable? Simulate a spatial migration of species in a Lotka-Volterra model. Use 1d diffision in a ring.
2.3. Investigate and simulate epidemics models, model a vaccination effect http://en.wikipedia.org/wiki/Epidemic_model. Simulate epidemics of a desease. Write a model of spatial spread of an epidemics.
Additional sources
Equilibria, Stability of equilibria points
MATHEMATICAL MODELS OF SPECIES INTERACTIONS. IN TIME AND
SPACE. JON C. ALLEN
http://chesterrep.openrepository.com/cdr/bitstream/10034/118016/3/chapter%202.pdf
https://www.math.duke.edu//education/postcalc/
A more complex model:
A Mathematical Model of Three-Species Interactions in an
Aquatic Habitat by J. N. Ndam, J. P. Chollom, and T. G.
Kassem
http://www.hindawi.com/isrn/appmath/2012/391547/
Epidemic models: starting page for the search
http://en.wikipedia.org/wiki/Epidemic_model
Chaotic behavior
Designing Chaotic Models EDWARD N. LORENZ
http://journals.ametsoc.org/doi/pdf/10.1175/JAS3430.1
Chaos in a long-term experiment with a plankton community.
Beninca E, Huisman J, Heerkloss R, Johnk KD, Branco P, Van
Nes EH, Scheffer M, Ellner SP.
Mathematical models predict that species interactions such
as competition and predation can generate chaos. However,
experimental demonstrations of chaos in ecology are scarce
...
http://www.ncbi.nlm.nih.gov/pubmed/18273017
Traffic shock waves discussion:
Christopher Lustri (Oxford, UK) Continuum Modelling of
Traffic Flow 2010
http://people.maths.ox.ac.uk/lustri/Traffic.pdf
More elementary presentation:
Kurt Bryan (Rose-Hulman). Traffic Flow I, II
http://www.rose-hulman.edu/~bryan/lottamath/traffic1.pdf
http://www.rose-hulman.edu/~bryan/lottamath/traffic2.pdf
Engineering textbook exposition
L.H. Immers and S. Logghe. (KATHOLIEKE UNIVERSITEIT
LEUVEN) Traffic Flow Theory 2002
https://www.mech.kuleuven.be/cib/verkeer/dwn/H111part3.pdf
Textbooks:
Richard Haberman. Mathematical models . Mechanical
Vibrations, Population dynamics and Traffic Flow. SIAM
1998 Chapter 6.\
C.Dym. Principles of mathematical modeling. Elsevier
2004. Chapter 6. \
Projects
2. Model and simulate traffic on a road with entries (and exits). Regulate the entry density to avoid jams. Estimate the effect of regulation on the travel time.
3. Model and simulate traffic at a two-line road when one line is closed. How the speed limit before the traffic jam will affect the jam and the travel time.
Additional sources (numerics and modeling)
Working groups for the projects: (the scheme is suggested by Kyle Zortman)
Equilibrium and dynamics of time-variable multistable systems Equilibria of a damaged frame. Modeling of equilibrium states of a frame with a number of damageable rods
Project: Waves, collisions, and coherent
motion of assemblages.
Three models of nonlinear waves and
multiple equilibria