Introduction to Mathematical Modelling

Winter 2011 projects

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. |