MATH 3820
Introduction to Mathematical Modelling
Winter 2011 projects

Choose one of the following papers. Suggestions of problems to consider are given, but you are free to do whatever you want. You should explain the model (avoid paraphrasing the author(s), if you become familiar enough with the model, the explanation should come naturally). Conduct some numerical simulations.. Explain how you would extend the work (you can extend if you want).

And do not hesitate to come to see me for questions, if you need help with the math, the simulations..

Author(s) R.V. Canfield and L.E. Borgman
Title Some distributions of time to failure for reliability applications
Details Technometrics, Vol. 17, No. 2 (1975)
Modelling method Probability
Remarks This paper is perfect is you want to explore in more detail the problem of time of residence in classes. Understand the approach (statistical in majority), but you can then try the same approach we used when we considered time of residence in classes: what happens when you use the different distributions in a cohort model?
Author(s) J.P. Eng
Title A mathematical model relating cohort and period mortality
Details Demography, Vol. 17, No. 1 (1980)
Modelling method Static model
Remarks This presents several functions that can be used to describe mortality. Understand the different functions used, study their shapes as a function of parameters, and try your hands with the data provided. There is a sort of discrete-time model hidden in this paper, as well, try to investigate it a little.
Author(s) R.A. Fisher
Title The wave of advance of advantageous genes
Details Annals of Eugenics, Vol. 7 (1937)
Modelling method Partial differential equations
Remarks This is an extremely well known paper of mathematical biology. Explain how the equation (1) is derived in a little more detail than is given here. Then reproduce the analysis of Fisher. Then perform numerical simulations (this should prove much easier than it was for Fisher).
Author(s) B.J. Hammond and D.A.J. Tyrrell
Title A mathematical model of common-cold epidemics on Tristan de Cunha
Details The Journal of Hygiene, Vol. 69, No. 3 (1971)
Modelling method Ordinary differential equations
Remarks This is a straightforward application of the SIR epidemic model of Kermack and McKendrick. Try to conduct the analysis, identify parameters, etc.
Author(s) S.B. Hsu, S.P. Hubbell and P. Waltman
Title A contribution to the theory of competing predators
Details Ecological Monographs, Vol. 48, No. 3 (1978)
Modelling method Ordinary differential equations
Remarks A classic of the chemostat literature, and more generally, a classical problem in ecology: suppose that two species are competing for the same nutrient. Is it possible for the two species to coexist?
Author(s) Y.A. Ilinskii, B. Lipkens, T.S. Lucas, T.W. Van Doren and E.A. Zabolotskaya
Title Nonlinear standing waves in an acoustical resonator
Details Journal of the Acoustical Society of America, Vol. 104 (1998)
Modelling method Partial differential equations
Remarks Study of a wave-type solution in an acoustic device. Understand the model, and see how the wave solutions are obtained. Try some numerics.
Author(s) A. Inselberg
Title Cochlear dynamics: the evolution of a mathematical model
Details SIAM Review, Vol. 20, No. 2 (1978)
Modelling method Partial differential equations
Remarks For this project, see also the following two articles, on which this one is based: Inselberg and Chadwick (1976) and Chadwick, Inselberg and Johnson (1976). This is a nice illustration of the progression of a model, from "simple" to more complicated. Using this paper and the two papers cited, explain the derivation of the model(s) and the analysis.
Author(s) K. Lange and F.J. Oyarzun
Title The attractiveness of the Droop equations
Details Mathematical Biosciences, Vol. 111 (1992)
Modelling method Ordinary differential equations
Remarks Well known paper in the chemostat field, which establishes global properties of the Droop model, a slightly more complicated model than the one in the notes about the chemostat, which accounts for an additional compartment, the cell quota. Establishing global stability is not easy, as can be inferred from the analysis here. Reproduce the local analysis, and see if you can understand the global one. Do the work both in nondimensional and in dimensional form. Perform some numerical simulations.
Author(s) R. Levins and D. Culver
Title Regional coexistence of species and competition between rare species
Details Proceedings of the National Academy of Sciences of the United States of America, Vol. 68, No. 6 (1971)
Modelling method Ordinary differential equations
Remarks A particular look at "space", using patches, i.e., interconnected discrete spatial units. Understand the model, run some numerics.
Author(s) G.B. Markus and P.E. Converse
Title A dynamic simultaneous equation model of electoral choice
Details The American Political Science Review, Vol. 73, No. 4 (1979)
Modelling method Discrete-time system
Remarks How the voting process could be explained, as a function of the successive statements of candidates. Formulate the model, run some numerics.
Author(s) R.M. May
Title Simple mathematical models with very complicated dynamics
Details Nature, Vol. 261 (1976)
Modelling method Discrete-time equations
Remarks This is a well known article that more or less started the whole "chaos buzz". Reproduce the results, develop the analysis. Beware: there are some errors in this paper (for example, with figures). See if you can spot them.
Author(s) C.J. Mode
Title A mathematical model for the co-evolution of obligate parasites and their hosts
Details Evolution, Vol. 12, No. 2 (1958)
Modelling method Ordinary differential equations
Remarks A classic paper in mathematical biology. A simple model about which plenty can be done: mathematical analysis, numerical simulations, etc.
Author(s) B.E. Newling
Title Urban growth and spatial structure: mathematical models and empirical evidence
Details Geographical Review, Vol. 56, No. 2 (1966)
Modelling method Static model
Remarks Describes the spread of a city and the population density. Could be used as a starting point to consider more detailed models, or could be used with more recent data.
Author(s) C.R. Rao and A.M. Kshirsagar
Title A semi-Markovian model for predator-prey interactions
Details Biometrics, Vol. 34, No. 4 (1978)
Modelling method Discrete-time Markov chains
Remarks This is a model that has a Markov chain model as a particular case, if the distribution of time spent in the different states is constant. Otherwise, it is only semi-Markovian.
Author(s) D.A. Roff
Title The analysis of a population model demonstrating the importance of dispersal in a heterogeneous environment
Details Oecologia, Vol. 15, No. 3 (1974)
Modelling method Discrete-time systems
Remarks A discrete-time system to consider the effect of various hypotheses about migration between location. Effect of stochasticity.
Author(s) J.A. Sherratt
Title Traveling wave solutions of a mathematical model for tumor encapsulation
Details SIAM Journal on Applied Mathematics, Vol. 60, No. 2 (1999)
Modelling method Partial differential equations
Remarks Not easy mathematically, but describes well the steps of the analysis, from how to establish the existence of travelling wave to the numerical treatment of the problem.
Author(s) W. Siler
Title A competing-risk model for animal mortality
Details Ecology, Vol. 60, No. 4 (1979)
Modelling method Probability
Remarks The problem of time of residence in classes is investigated (with focus on mortality). Competing risks consider the different hazards that an individual is exposed to, the hazard is the derivative of the survival function.
Author(s) J.W. Sinko and W. Streifer
Title A model for population reproducing by fission
Details Ecology, Vol. 52, No. 2 (1971)
Modelling method Partial differential equations
Remarks Considers a population of cells that reproduce by fission. Understand the model, the analysis ("solving along characteristics"), and do numerical simulations.
Author(s) M. Slatkin
Title Competition and regional coexistence
Details Ecology, Vol. 55, No. 1 (1974)
Modelling method Ordinary differential equations
Remarks A model for the competition of two species living in two different locations. Derive the model, perform the analysis. Then do the numerics, which were done at the time using a very simple approximation method.
Author(s) R.A.J. Taylor
Title A simulation model of locust migratory behaviour
Details The Journal of Animal Ecology, Vol. 48, No. 2 (1979)
Modelling method Computer simulations
Remarks This paper presents a computer simulation model of locust migratory behaviour. Write a discrete-time mathematical model that could be used to run the simulations. (Simplify the problem if needed.)
Author(s) L.E. Thomas
Title A model for drug concentration
Details SIAM Review, Vol. 19, No. 4 (1977)
Modelling method Impulsive differential equation (ordinary differential equation with discrete-time events)
Remarks This is an equation of a type we will not study in class, an impulsive equation. This is an ODE which is subject to periodic forcing (here, to model the injection of drugs). This results in a discrete-time equation. Understand the model, the analysis. Conduct numerical analysis (there is none on the paper), it is easy to find parameters of drugs commonly used.
Author(s) M.B. Usher
Title A matrix model for forest management
Details Biometrics, Vol. 25, No. 2 (1969)
Modelling method Discrete-time systems
Remarks A discrete-time model for a forest, with planting, harvesting, etc. This is an example of a discrete-time model with structure, of the same nature as the well known Leslie matrix model.
Author(s) J. Yellin and P.A. Samuelson
Title A dynamical model for human population
Details Proceedings of the National Academy of Sciences, Vol. 71, No 7 (1974)
Modelling method Ordinary differential equations
Remarks This is a two sex model for a population. Go through the analysis, in particular, show how the table on page 2816 is obtained. Perform some numerical simulations.