The Final Bugbox Model
Let Lt be the number of larvae at time t.
Let Pt be the number of pupae at time t.
Let At be the number of adults at time t.
Lt+1 = sLt
+ fAt
Pt+1 = pLt
At+1 =
Pt + aAt
Discovering Model Behavior
• We now guide the students through a
process of discovery:
– The students write Matlab code to run
simulations with given parameters and initial
data.
– We direct the students to the discovery that
the proportions always approach a fixed ratio
by asking them to plot P/A. We also have
them plot Xt+1/Xt for each stage.
Model Behavior
• We fix p = a = 0.5, s = 0.25, f = 2, and choose
initial population levels. Plot r = P/A.
4
3
P/A
2
1
0
0
5
10
15
t
The ratio r approaches a constant that
is independent of the initial conditions.
Analysis of the Model
• The stable growth rate is the eigenvalue of
largest magnitude. The stable population
ratios are given by the corresponding
eigenvector.
• The students find the stable growth rate
and population ratios using only basic
algebra and “directed ingenuity.”
Analysis of the Model
• “P/A approaches a constant” means that P
and A have the same growth rate as t →∞.
Assume L, P, and A all have growth rate λ.
Lt+1 = λ Lt , Pt+1 = λ Pt , At+1 = λ At .
λ Lt = ¼ Lt
+ 2At
λ Pt = ½ Lt
λ At =
Pt + ½ At
λ Lt = ¼ Lt
+ 2At
λ Pt = ½ Lt
λ At =
Pt + ½ At
Take Lt = 2λ Pt (equation 2)
Then
2λ(λ - ¼) Pt = 2At , Pt = (λ - ½) At
(eqs 1,3)
λ Lt = ¼ Lt
+ 2At
λ Pt = ½ Lt
λ At =
Pt + ½ At
Take Lt = 2λ Pt (equation 2)
Then
2λ(λ - ¼) Pt = 2At , Pt = (λ - ½) At
1
r = ―――― ,
λ(λ-¼)
r = λ-½
(eqs 1,3)
(r = P/A)
λ Lt = ¼ Lt
+ 2At
λ Pt = ½ Lt
λ At =
Pt + ½ At
Take Lt = 2λ Pt (equation 2)
Then
2λ(λ - ¼) Pt = 2At , Pt = (λ - ½) At
1
r = ―――― ,
λ(λ-¼)
r = λ-½
λ(λ-¼)(λ-½)=1
(eqs 1,3)
(r = P/A)
A constant ratio is possible only for special
values of λ !
λ(λ-¼)(λ-½)=1
r = λ-½
A constant ratio is possible only for special
values of λ !
λ(λ-¼)(λ-½)=1
r = λ-½
1.5
λ ≈ 1.27
1
0.5
0
0
0.5
1

r ≈ 0.77
1.5
Similarly, we get L/A ≈ 1.96.
The growth rate is ≈ 1.27, and
the population ratio is ≈ 1.96:0.77:1.
Mathematical Questions
• Can we improve the calculation method
by using a more sophisticated
mathematical structure?
Mathematical Questions
• Can we improve the calculation method
by using a more sophisticated
mathematical structure?
• What happens if we allow the
parameters to take on any biologicallyrealistic values?
Mathematical Questions
• Can we improve the calculation method
by using a more sophisticated
mathematical structure?
• What happens if we allow the
parameters to take on any biologicallyrealistic values?
– Are solutions always positive?
– Do simulations always tend toward stable
ratio and steady growth rate?
– Can there be more than one solution?
– If so, what do the simulations actually do?
Improving the Method
• Introduce matrix notation.
• Redo the calculation using matrix notation.
– Introduce eigenvalues and eigenvectors.
• Develop the 2-step solution method:
– det (M-λI) = 0 to find the eigenvalues.
– (M-λI) x = 0 to find the eigenvectors.
Building the Theory
• We can prove there are no more than n
eigenvalues.
Building the Theory
• We can prove there are no more than n
eigenvalues.
• We can prove that long-term behavior is
determined by the eigenvalue of largest
modulus.
Building the Theory
• We can prove there are no more than n
eigenvalues.
• We can prove that long-term behavior is
determined by the eigenvalue of largest
modulus.
• We can state an existence theorem:
– Nonnegative, irreducible, primitive matrices
have a unique eigenvalue of largest modulus.
This eigenvalue is real and its eigenvector is
real and strictly positive.
Results of Student Work
• We got excellent lab data to determine
model parameters.
• Population growth data was consistent
with model predictions, but with large
experimental uncertainty.
• Predator-prey data was not useful, but we
still learned the theory and mathematics.
Assessment -- Surveys
• Big positive changes from pre- to post– If I need to, I can write a short computer
program to study a mathematical model.
– A lot of theoretical biology is mathematical.
– I can learn a lot of biology from reading
articles and books.
• No negative changes