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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:
NSF support for this MiniCourse is gratefully acknowledged! |
Announcement (http://www.ams.org/amsmtgs/2098_minicourses.html):
SOLD OUT Minicourse #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
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Discrete Population Models
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Three 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
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Chemostat Models
.pdf
.ppt
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Dynamics 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.
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Physiologically Based Pharmacokinetic (PBPK) Models
.pdf
.ppt
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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.
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
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Hodgkin-Huxley and Fitzhugh-Nagumo Models
.pdf
.ppt
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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.
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