Teaching Biology in Mathematics Classes

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Teaching Biology in Mathematics Classes

Glenn Ledder Department of Mathematics University
of Nebraska-Lincoln gledder_at_math.unl.edu funded
by NSF grant DUE 0531920
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Overview
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

1. Comar, PRIMUS 18, 49-70, 2008
2. Ledder, PRIMUS 18, 119-138, 2008
3. Ledder, Differential Equations A Modeling
   Approach, McGraw-Hill, 2005
4. Ledder, Mathematical Methods for Biology and
   Medicine, in preparation

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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?

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Energy Budget Modeling

• Introducing the kyoob, a biologically simple
  creature of cubic shape
• Intake rate is proportional to surface area 6as2
• Use for tissue maintenance is proportional to
  volume bs3
• Surplus resources are used for growth to size S
  and then reproduction.
• Kyoobs live forever.
• Goals Find the physiological maximum size and
  the optimal adult size.

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Energy Budget Modeling

• Surplus Energy
• Physiological Maximum
• (no surplus)
• Optimal Size

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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?

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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.

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• Let p by the prevalence of A.
• Let q1-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
Frequency p2 2pq q2
Product (1-m) p2 2pq 0
w(p)(1-m) p2 2p(1-p)
1 1m
Optimum p is .
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• 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 qt1
  from qt when m0. ANSWER qt1qt/(1qt)
• Solve the difference equation to obtain a
  sequence for q. ANSWER qt1/(t5)
• How many generations does it take to reduce
  sickle cell deaths to 1 in 10,000? ANSWER 95

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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?

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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?
• The birth rate should increase exponentially with
  rate r.
• The birth rate can be computed by adding up the
  births to parents of different ages.

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Demographics / Population Growth

• Population of age x if no deaths
• Actual population of age x
• Birth rate for parents of age x
• Total birth rate at time t
• Total birth rate at time t
• Euler equation

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• 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.

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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?

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• Presenting Bugbox-population, a biology lab for a
  virtual world.
• http//www.math.unl.edu/gledder1/BUGBOX/
• Boxbugs are simpler than real insects
• They dont 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.

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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.

Lt1 s Lt f At
Pt1 p Lt
At1 Pt a At
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• Things to do with the model
• Write as xt1 M xt .
• Run a simulation to see that x evolves to a fixed
  ratio independent of initial conditions.
• Obtain the problem M xt ? xt .
• Develop eigenvalues and eigenvectors.
• Show that the term with largest ? 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.

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Computer Simulation Results
A plot of Xt/Xt-1 shows that all variables tend
to a constant growth rate ?
The ratios LtAt and PtAt tend to constant
values.
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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?

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Pharmacokinetics
Q(t)
k1 x
blood
tissues
k2 y
x(t)
y(t)
r x

• x' Q(t) (k1r) x k2 y
• y' k1 x k2 y

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• 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, r1, k11. What
  happens for different choices of k2lt1?
• Relate the result to lead poisoning.

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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?

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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

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Predator-Prey Dynamics

• Lotka-Volterra
• x prey, y predator
• x' r x s x y
• y' e s x y m y
• Predicts oscillations of varying amplitude

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Predator-Prey Dynamics

• Lotka-Volterra
• x prey, y predator
• x' r x s x y
• y' e s x y m y
• Predicts oscillations of varying amplitude
• Predicts impossibility of predator extinction.

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Predator-Prey Dynamics

• logistic
• x prey, y predator
• x' r x (1 ) s x y
• y' e s x y m y
• Predicts stable x y equilibrium if m is small
  enough

x K
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Predator-Prey Dynamics

• logistic
• x prey, y predator
• x' r x (1 ) s x y
• y' e s x y m y
• Predicts stable x y equilibrium if m is small
  enough and y?0 if m too large

x K
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Predator-Prey Dynamics

• Holling type 2
• x prey, y predator
• x' r x (1 )
• y' m y

x K
q x y A x
e q x y A x
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q x y A x

• Why ?
• 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 f s x and f h P 1
• P

s x 1 sh x
q x A x
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Predator-Prey Dynamics

• Holling type 2
• x prey, y predator
• x' r x (1 )
• y' m y
• Predicts stable x y equilibrium if m is small
  enough.

x K
q x y A x
e q x y A x
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Predator-Prey Dynamics

• Holling type 2
• x prey, y predator
• x' r x (1 )
• y' m y
• Predicts stable x y equilibrium if m is small
  enough and stable limit cycle if m is even
  smaller.

x K
q x y A x
e q x y A x
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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?

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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.

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• Holling type 3 consumption
• Saturation and alternative resource

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Dimensionless Version

• k represents the environmental capacity.
• c represents the number of consumers.
• Decreasing A increases both k and c.

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The resource increases
The resource decreases
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Stage 1 natural balance
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Stage 2 depletion
Consumption increases to high level.
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Stage 3 inadequate correction
Consumption decreases to modest level.
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Stage 4 recovery
Consumption decreases to minimal level.
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Stage 5 proper management
Consumption increases to modest level.
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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, Whats 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