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## Some Deterministic Models in Mathematical Biology and Their Simulations using Phaser AMS-MAA National Meeting in New Orleans, January 05-07, 2007

Organizers:

- Cammey Cole (Meredith College): Organizer and speaker
- Brian Coomes (University of Miami): Collaborator
- Jason Glick (Phaser Scientific Software LLC): Principal software engineer
- Huseyin Kocak (University of Miami): Organizer and speaker
- Craig Kolthoff (University of Miami): Network and Web services
- Burton Rosenberg (University of Miami): Collaborator
- James Selgrade (North Carolina State University): Organizer and speaker
NSF support for this MiniCourse is gratefully acknowledged!

## Announcement (http://www.ams.org/amsmtgs/2098_minicourses.html):

SOLD OUTMinicourse #2 Some deterministic models in mathematical biology and their simulations, organized by James F. Selgrade, North Carolina State University, Cammey E. Cole, Meredith College, and Hüseyin Koçak, University of Miami, Coral Gables; Part A: Friday, 2:15 p.m. to 4:15 p.m. and Part B: Sunday, 1:00 p.m. to 3:00 p.m. This 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. Each of the four topics will be discussed for 30 minutes followed by 30 minutes of computer experimentation. The participants will be provided electronic copies of the Web-based notes, simulations, and the software. Familiarity with the material in undergraduate courses in ordinary differential equations and linear algebra will be helpful. Cost is US$95; enrollment limit is 30.

Note:Majority of the Phaser simulations (.ppf and .pgf files) in the lectures below require Phaser 3.0 which will be released towards the end of January 2007. Pre-release copies of Phaser 3.0 will be made available to the participants at the Mini Course.

## THE LECTURES

Discrete Population ModelsThree examples of population dynamics models framed in terms of difference equations will be simulated:

- Ricker MAP: A pioneering model of a density-dependent population
- Nicholson-Bailey MAP: A pioneering model of host-parasitoid interactions
- Beddington-Free-Lawton MAP: Stabilizing Nicholson-Bailey with host density dependence
Speaker: Huseyin Kocak, University of Miami.

Chemostat Models .pdf .pptDynamics of microbial competition and predator-prey in cultures.

Speaker: James Selgrade, North Carolina State University.

Phaser Simulations:

chemo1.ppf chemo2.ppf chemo2.pgf chemo3.ppf chemoeq4.pgf 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.

Physiologically Based Pharmacokinetic (PBPK) Models .pdf .pptA 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.

Speaker: Cammey Cole, Meredith College.

Phaser Simulations:LinearModel.ppf Variables vs.time of the linear model

LinearModel_a.pgf A Slideshow of the linear model as the parameter a is varied

NonLinearModel.ppf Variables vs.time of the non-linear model

NonlinearChanging_a.pgf A Slideshow of the non-linear model as the parameter a is varied

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

Speaker: James Selgrade, North Carolina State University.

Phaser Simulations:Hudgkin-Huxley-AP.pgf A Slideshow of the action potential in the Hudgkin-Huxley equations as the parameter I is varied

Hudgkin-Huxley-Hopf.ppf Bifurcation diagram depicting a Poincare-Andronov-Hopf bifurcation in the Hudgkin-Huxley equations

FN-stablefocus.ppf A stable focus in the Fitzhugh-Nagumo equations

FN-smallcycle.ppf Small amplitude oscillations in the Fitzhugh-Nagumo equations

FN-stablecycle.ppf Large amplitude oscillations in the Fitzhugh-Nagumo equations

FN-orbitshow.pgf A Slideshow of growing oscillations in the Fitzhugh-Nagumo equations as the parameter I is varied

Fitzhugh-Nagumo.pgf A slideshow of voltage outputs as the values of the input current is changed in the Fitzhugh-Nagumo equations

Fitzhugh-Nagumo-Hopf.ppf Bifurcation diagram depicting a Poincare-Andronov-Hopf bifurcation in the Fitzhugh-Nagumo equations