Mathematics in Biology??!!
Mark Pernarowski
Montana State University, Bozeman
What Kind of Biology?








Population of organisms - i.e., animals
Spread of Disease - i.e., Flu, vaccines,
How muscles work - even the heart
Diabetes - insulin and glucose
Patterns in natures - zebra stripes, leopard spots
Brain electrical behavior - sight, sound
Organism growth - embryos
Genetics …..and much more
Population Problems
 Predator-Prey
 X=hare population
 Y=lynx population
Spread of Disease
 Hong Kong Flu (1968-69 NYC).
 Pre-vacinne flu related pneumonia
SIR Model
 Susceptibles S
 Infected I
 Recovered R
How do leeches swim?




Physics of Fluids
F(x,t)=leech force
u=fluid velocity
p=pressure
QuickTime™ and a
Animation decompressor
are needed to see this picture.
Patterns in Nature
 Chemicals that react
and diffuse in animal
coats make visible
patterns
 c(x,t) concentration at
time t location x.
Embryo Development -genes
 Young embryos form a body axis early on
 Why? How? Chemicals cause cells to move.
Models Reproduce
Experiments
 n=cell density
 u=chemical
Some Biology is “Stimulus/Response”
 A stimulus I causes an output u
 Sometimes get an output even if there is no stimulus (I=0)
(i.e., people who talk to you even when you don’t talk to them)
Lot’s of different kinds of math models:
Visual System
 The electrical
firing rate of
your visual
system neurons
changes with
stimulus angle!
Each neuron can be modeled.
 Ions move across
membrane
 Voltage V
changes in time
Models of Visual System Orientation Patterns
 Activity a(r,t) at time t, position r in your brain.
 On - neurons fire
immediately
 Off- neurons fire
after light turned
off
 Math models
explain both and
more.
Diabetes
 An increase in glucose (sugar) causes pancreas cells to
make insulin. Calcium plays a key role.
Bursting Electrical Activity
 Calcium makes cells voltage “burst”
 Same kind of math models as neurons.
World’s smallest engine!
 Molecules which make “ATP” (your energy)
QuickTime™ and a
Video decompressor
are needed to see this picture.
A simpler math model of animal populations

is population at year n=1,2,3,….
 Less animals next year if population is too high. Why?
Maybe population levels out
http://www.math.montana.edu/frankw/ccp/modeling/discrete/logistic/appwindow.htm
Maybe population alternates (period 2)
Maybe population alternates (period 4)
Maybe population chaotic
Conclusion
 There will be more math in biology and
medicine in your lifetime.
 Unlike other science the math models are
being discovered right now!