Teaching Biology in
Mathematics Classes
Glenn Ledder
Department of Mathematics
University of Nebraska-Lincoln
gledder@math.unl.edu
funded by NSF grant DUE 0531920
Overview
1.
2.
3.
4.
Biology Topic
Mathematics Topic
Energy budget modeling4
Single Variable Optimization
Genetics and evolution1
Sequences and difference equations
Demographics and population growth1
Integration
Structured population dynamics2
Matrix multiplication and eigenvalues
Pharmacokinetics3
Linear systems of ODEs
Predator-prey dynamics4
Nonlinear systems of ODEs
Resource management4
Nonlinear first-order ODEs
Comar, PRIMUS 18, 49-70, 2008
Ledder, PRIMUS 18, 119-138, 2008
Ledder, Differential Equations: A Modeling Approach, McGraw-Hill, 2005
Ledder, Mathematical Methods for Biology and Medicine, in preparation
Energy Budget Modeling
• What do organisms do with the resources
they collect from their food?
• Why do different species grow to different
sizes?
• Do organisms grow to their physiological
maximum size?
Energy Budget Modeling
Introducing the “kyoob,” a biologically simple
creature of cubic shape:
1. Intake rate is proportional to surface area: 6as2
2. Use for tissue maintenance is proportional to
volume: bs3
3. Surplus resources are used for growth to size S
and then reproduction.
4. Kyoobs live forever.
Goals: Find the physiological maximum size
and the optimal adult size.
Energy Budget Modeling
Surplus Energy:
R(s)  6as  bs
2
Physiological Maximum:
(no surplus)
Optimal Size:
smax
3
6a

b
0  R' (S )  12aS  3bS
4a 2
S
 smax
b 3
2
Genetics and Evolution
• How does natural selection change the
gene pool?
• Why did natural selection favor the gene
for sickle cell anemia?
• Should sickle cell anemia disappear in the
future? How quickly?
Genetics and Evolution
Sickle cell anemia biology:
• Everyone has a pair of genes (each either
A or a) at the sickle cell locus:
– AA: vulnerable to malaria
– Aa: protected from malaria
– aa: sickle cell anemia
• Babies get A from an AA parent and
either A or a from an Aa parent.
Let p by the prevalence of A.
Let q=1-p be the prevalence of a.
Let m be the malaria mortality.
Let w(p) be a measure of the relative fitness of the
gene pool.
Genotype
AA
Aa
aa
Fitness
1-m
1
0
p2
2pq
q2
(1-m) p2
2pq
0
Frequency
Product
w(p)=(1-m) p2 + 2p(1-p)
1
Optimum p is ——–.
1+m
Things to do with the model:
• Explore the action of natural selection when
modern medicine changes m to 0.
– Find the value q0 that yields a sickle cell incidence
rate of 4%. [ANSWER: 0.2]
•
.
– Find the corresponding value m0. [ANSWER: 0.25]
– Derive a difference equation that determines qt+1 from
qt when m=0. [ANSWER: qt+1=qt/(1+qt)]
– Solve the difference equation to obtain a sequence
for q. [ANSWER: qt=1/(t+5)]
– How many generations does it take to reduce sickle
cell deaths to 1 in 10,000? [ANSWER: 95]
Demographics / Population Growth
• What determines the rate of growth of a
population?
• How are the ages of members of a
population distributed?
• In particular, how do changes in birth rates
and mortality rates change population
growth and structure?
Demographics / Population Growth
Let l(x) be the probability of survival to age x.
Let m(x) be the rate of production of offspring for
parents of age x.
Let r be the population growth rate.
Let B(t) be the total birth rate.
How do l and m determine B and r?
1. The birth rate should increase exponentially with
rate r.
2. The birth rate can be computed by adding up the
births to parents of different ages.
Demographics / Population Growth
B(t  x) dx
Population of age x if no deaths:
B(t  x)l ( x) dx
Actual population of age x:
Birth rate for parents of age x: B(t  x)l ( x)m( x) dx
Total birth rate at time t:

B(t )   B(t  x)l ( x)m( x) dx
0
B(t )  B(0) e rt
Total birth rate at time t:
Euler equation:

1  e
0
 rx
l ( x)m( x) dx
Things to do with the model:
• Discretize it by assuming that l and m are
piecewise constant (integrating over each time
interval.
• Find real data for l and m. Then use a numerical
solver to find r.
• Explore the changes in r when
–
–
–
–
Births are delayed
Parents have fewer babies
Mortality decreases for the elderly
Infant mortality decreases.
Structured Population Dynamics
• Can we create a simple model that tracks
changes in the age/size/stage distribution
of a population?
• How do development rates affect
population growth?
Presenting Bugbox-population, a biology lab
for a virtual world.
http://www.math.unl.edu/~gledder1/BUGBOX/
Boxbugs are simpler than real insects:
– They don’t move.
– Development rate is chosen by the experimenter.
– Each life stage has a distinctive appearance.
larva
pupa
adult
• Boxbugs progress from larva to pupa to adult.
• All boxbugs are female.
• Larva are born adjacent to their mother.
Structured Population Dynamics
The final “bugbox” model:
Let Lt be the number of larvae at time t.
Let Pt be the number of juveniles at time t.
Let At be the number of adults at time t.
Lt+1 = sLt
+ fAt
Pt+1 = pLt
At+1 =
Pt + aAt
Things to do with the model:
• Write as xt+1 = Mxt .
• Run a simulation to see that x evolves to a fixed
ratio independent of initial conditions.
• Obtain the problem Mxt = λxt .
• Develop eigenvalues and eigenvectors.
• Show that the term with largest |λ| dominates
and note that the largest eigenvalue is always
positive.
• Note the significance of the largest eigenvalue.
• Use it to predict long-term behavior and discuss
its shortcomings.
Computer Simulation Results
A plot of Xt/Xt-1 shows
that all variables tend to
a constant growth rate λ
The ratios Lt:At
and Pt:At tend to
constant values.
Pharmacokinetics
• How long does it take before IV
medication takes effect?
• Why do we have to take some medication
once a day and other medication every
four hours?
• Why does food poisoning last only a short
time but lead poisoning lasts forever?
Pharmacokinetics
Q(t)
blood
x(t)
k1 x
k2 y
tissues
y(t)
rx
x′ = Q(t) – (k1+r) x + k2 y
y′ = k1 x – k2 y
Things to do with the model:
• Find equilibrium solutions when Q is constant.
• Show that the eigenvalues are both negative.
• Find realistic parameter values for some
medication and run simulations.
• Take x(0)=0, y(0)=1, Q(t)=0, r=1, k1=1. What
happens for different choices of k2<1?
– Relate the result to lead poisoning.
Predator-Prey Dynamics
• How do the interactions between
predators and prey affect the populations
of both?
• How does selective killing of predators
change the predator and prey
populations?
Predator-Prey Dynamics
• General idea
x = prey (biomass), y = predator (biomass)
x′ = growth rate without predators
- loss due to predation
y′ = growth rate from predation
- loss rate without prey
Predator-Prey Dynamics
• Lotka-Volterra
x = prey, y = predator
x′ = rx – sxy
y′ = esxy – my
Predicts oscillations of
varying amplitude
Predator-Prey Dynamics
• Lotka-Volterra
x = prey, y = predator
x′ = rx – sxy
y′ = esxy – my
Predicts oscillations of
varying amplitude
Predicts impossibility of
predator extinction.
Predator-Prey Dynamics
• logistic
x = prey, y = predator
x
x′ = rx 1 – —
K
y′ = esxy – my
(
) – sxy
Predicts stable xy
equilibrium if m is
small enough
Predator-Prey Dynamics
• logistic
x = prey, y = predator
x
x′ = rx 1 – —
K
y′ = esxy – my
(
) – sxy
Predicts stable xy
equilibrium if m is
small enough and
y→0 if m too large
Predator-Prey Dynamics
• Holling type 2
x = prey, y = predator
x
qxy
x′ = rx 1 – —
–
—–––
K
A+ x
eqxy
y′ = —–––
–
my
A+ x
(
)
qxy
Why —–––
?
A+ x
Let s be search rate
Let P be predation rate per predator
Let f be fraction of time spent searching
Let h be the time needed to handle one prey
P = fsx and f + hP = 1
sx
qx
P = —–––– = —–––
1 + shx
A+ x
Predator-Prey Dynamics
• Holling type 2
x = prey, y = predator
x
qxy
x′ = rx 1 – —
–
—–––
K
A+ x
eqxy
y′ = —–––
– my
A+ x
(
)
Predicts stable xy
equilibrium if m is small
enough.
Predator-Prey Dynamics
• Holling type 2
x = prey, y = predator
x
qxy
x′ = rx 1 – —
–
—–––
K
A+ x
eqxy
y′ = —–––
– my
A+ x
(
)
Predicts stable xy
equilibrium if m is small
enough and stable limit
cycle if m is even smaller.
Resource Management
• Why have natural resources, such as
whales or bison, been depleted so
quickly?
• How can we restore natural resources?
• How should we manage natural
resources?
Resource Management
Let X be the biomass of resources.
Let K be the environmental capacity.
Let C be the number of consumers.
Let G(X) be the consumption per consumer.
dX
 X
 R X 1    C G ( X )
dT
 K
• Holling type 3 consumption
– Saturation and alternative resource
2
QX
G( X )  2
A X2
Q
0.75Q
G 0.5Q
0.25Q
0
0
A
2A
X
3A
4A
Dimensionless Version
t
K
CQ
X  Ax, T  , k  , c 
R
A
RA
1  x 
dx
x 
 cx  1   
2
dt
c  k  1 x 
k represents the environmental capacity.
c represents the number of consumers.
Decreasing A increases both k and c.
1  x 
dx
x 
 c x  1   
2
dt
c  k  1 x 
1 x
x
1   
2
c  k  1 x
The resource increases
x
1 x
 1  
2
1 x
c k
The resource decreases
Stage 1 – natural balance
x
Stage 2 – depletion
Consumption increases to high level.
x
Stage 3 – inadequate correction
Consumption decreases to modest level.
x
Stage 4 – recovery
Consumption decreases to minimal level.
x
Stage 5 – proper management
Consumption increases to modest level.
x
PRIMUS 18(1), 2008
• Teaching Math to Biology Students:
– J.P. Fulton and L. Sabatino, Using the scientific method to
motivate biology students to study precalculus
– J.D. White and J.P. Carpenter, Integrating mathematics into the
introductory biology laboratory course
– R.H. Lock and P.F. Lock, Introducing statistical inference to
biology students through bootstrapping and randomization
• Teaching Biology to Math Students:
– T.D. Comar, The integration of biology into calculus courses
– R. Burks, J. Lindquist, S. McMurran, What’s my math course got
to do with biology?
– E. Marland, K.M. Palmer, R.A. Salinas, Biological applications in
the mathematics curriculum
– L.J. Heyer, A mathematical optimization problem in bioinformatics
• Mathematical Modeling:
– G. Ledder, An experimental approach to mathematical modeling
in biology