Some Deterministic Models in Mathematical Biology,
A Workshop for the Mathematical Association of America

Organizers: James Selgrade (North Carolina State University), Cammey Cole (Meredith College, and Huseyin Kocak (University of Miami)

This short course will present and analyze discrete and continuous models from physiology (e.g., the Hodgkin-Huxley model), pharmacokinetics, and population biology (e.g., the chemostat model). The class will be conducted in a computer lab where participants will use the software Phaser to simulate model behavior. Familiarity with the material in undergraduate courses is ordinary differential equations and linear algebra will be helpful.

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Physiologically Based Pharmacokinetic (PBPK) Models    .pdf     .ppt

A physiologically based pharmacokinetic (PBPK) model for the uptake and elimination of a chemical in rodents is developed to relate the amount of IV and orally administered chemical to the tissue doses of the chemical and its metabolite.
Author: Cammey Cole, Meredith College.

Phaser Simulations:
LinearModel.ppf
NonLinearModel.ppf

Discrete Population Models    .pdf     .ppt

Dynamics of cooperative, competitive, and population genetics models using difference equations are studied.
Author: James Selgrade, North Carolina State University.

Phaser Simulations:
Logistic.pgf A Gallery of stairstep diagrams of the Logistic Map.
LogisticBifurcation.ppf Bifurcation diagram of the Logistic Map.
Genetics.pgf A Gallery of phase portraits of Genetics MAP.

Hodgkin-Huxley and Fitzhugh-Nagumo Models    .pdf     .ppt

A brief dicussion of the physiology of the action potential of excitable cells is presented. Then the celebrated model Hodgkin-Huxley is introduced and its dynamics is illustrated with numerical simulations. Lastly, the simplification of this model due to Fitzhugh-Nagumo is analyzed.
Author: James Selgrade, North Carolina State University.

Phaser Simulations:
Hudgkin-Huxley-AP.pgf Action potential in the Hudgkin-Huxley equations.
Hudgkin-Huxley-Hopf.ppf Bifurcation diagram depicting a Poincare-Andronov-Hopf bifurcation in the Hudgkin-Huxley equations.
Fitzhugh-Nagumo.pgf A gallery of voltage outputs for several values of the input current in the Fitzhugh-Nagumo equations.
Fitzhugh-Nagumo-Hopf.ppf Bifurcation diagram depicting a Poincare-Andronov-Hopf bifurcation in the Fitzhugh-Nagumo equations.

Chemostat Models    .pdf     .ppt

Dynamics of microbial competition and predator-prey in cultures.
Author: James Selgrade, North Carolina State University.

Phaser Simulations:
chemo1.ppf chemo2.ppf chemo3.ppf chemo-3D.ppf Phase portraits of microbial competition.
chemostat-pp-3D.ppf A 3D view of limit cycle in microbial predator-prey.
chemostat-pp-Hopf.ppf Bifurcation diagram depicting a Poincare-Andronov-Hopf bifurcation in microbial predator-prey.