Integrated Quantitative Science
Long-term Behavior Homework
1. Text1 Section 2.5, Exercise 22.
2. Text Section 2.5, Exercise 28.
3. In a 1995 paper in the journal Science2, a team of ecologists reported the
following survivorship data for Clupea harengus, a species of herring:
S
10
40
70
100
130
160
190
220
250
280
310
340
370
400
430
460
490
R
800
1916
805
3190
1079
720
1197
3960
1175
2828
1643
1122
1073
1880
1032
3772
2499
Here, S = number of spawners (indep. var.) and R = number of “recruits”, i.e.
newborns that survive (dep. var.). The authors approximated the relationship
between S and R with a mathematical model known as the modified Beverton-Holt
model. This model is
419.35𝑆 0.56
𝑅 = 1+0.25𝑆0.56.
Throughout this assignment, all references to “our textbook” or “text” refer to the IQS calculus text:
Calculus: Concepts and Contexts, Stewart, 2nd edition.
2
Myers RA, Barrowman NJ, Hutchings JA, and Rosenberg AA. Population dynamics of exploited fish
stocks at low population levels, Science 269, 25 August 1995, 1106-1108. (Copy posted in Blackboard as
MyersEtAl1995.pdf.)
1
a. Use Excel to plot the data set and the model together in the same scatter
plot. Based on this plot, would you consider this to be a believable model?
Briefly justify your answer.
b. According to this model, what will happen to the number of recruits as the
number of spawners increases? Be as precise as you can (i.e. use the
model formula, not the scatter plot, to answer).
4. (continuation of Exercise 3) Ecologists (including the authors of the Science
article) are more interested in a quantity called recruitment yield, denoted Y,
𝑅
which is the number of recruits per spawner (i.e. 𝑌 = 𝑆 ). More precisely, they
want to know how the size of Y is related to the number of spawners.
a. Use Excel and the data from the preceding exercise to compute values of Y
that correspond to the values of S in the table. Make a scatter plot of Y
versus S.
𝑅
b. Using 𝑌 = 𝑆 and the model from Exercise 3, write a mathematical model
for the relationship between Y and S. Add this model to your scatter plot
from part a. Based on this plot, would you consider this to be a believable
model? Briefly justify your answer.
c. According to your model from part b., what will happen to the recruitment
yield as the number of spawners increases? Be as precise as you can.
Food Safety Research. Another important area of research which combines and
integrates ideas and tools from different scientific disciplines is the field of food
safety research. As an example, a 1993 paper3 reports a study of growth of the
bacteria Enterococcus faecium4 inside a type of Italian sausage called mortadella
bologna5.
The authors use the following mathematical model of the growth of E. faecium over
time inside a 32⁰C sausage:
3
Zanoni B, Garzaroli C, Anselm S, and Rondinini G, Modeling the growth of Enterococcus faecium in
bologna sausage, Applied and Environmental Microbiology 59 (10), Oct. 1993, 3411-3417. (Copy posted in
Blackboard as ZanoniEtAl1993.pdf.)
4
E. faecium is a gram-positive bacteria that lives harmlessly in the intestinal tract of many animals,
including humans. (Please don’t ask how human intestinal bacteria gets into bologna.) Sometimes, it
spreads to other parts of the body, and can cause dangerous, even deadly, infections, especially if the strain
is antibiotic resistant (as more than half now are, according to CDC estimates). In fact, vancomycinresistant E. faecium (VRE) is considered one of the “Big Three” most-problematic drug-resistant grampositives. (The other two are MRSA and resistant Strep.)
5
In English, mortadella bologna translates literally into “died of bologna”. (I am NOT making this up.)
Perhaps the name originally applied only to the sausages infected with VRE.
𝑃 = 𝑃0 𝑒 9.8901𝑒
−𝑒2.5420−0.2167𝑡
,
(Yes, it really is an “e to the e to the e” function!), where t is time in hours, P is the
number of bacteria at time t, and 𝑃0 is the initial number of bacteria (i.e. the number
of bacteria when t = 0).
5. For this bologna bacteria model:
a. Use a graph to guess at what this model predicts as the long-term size of
the bacteria population, if it starts with 100 bacteria.
b. Use a limit to determine the exact model prediction for part a.
More fun with bacteria. Way back in the first IQS lab (when we exposed E. coli to
varying levels of UV radiation, in an effort to mutate them), we assumed something about
bacterial growth that we will now stop and think about. We started with some test tubes
of E. coli, which Dr. Hill had grown overnight. We assumed that each of these test tubes
had an E. coli density of 108 bacteria per milliliter of solution, based on published studies
that say this is the carrying capacity of E. coli in broth. Essentially, we were assuming
that the initial E. coli population (whatever it was) had reached its carrying capacity
during the overnight period. Now, let’s investigate this, using the modeling and analysis
tools we have developed. Specifically, we will use the logistic growth IVP model
𝑑𝑃
𝑑𝑡
𝑃
= 𝑘𝑃 (1 − 𝐵) , 𝑃(0) = 𝐴,
where P is the density (in # bacteria/mL of solution) of E. coli, t is the time (in hours)
since the E. coli were added to the solution, B is the carrying capacity (also in #
bacteria/mL of solution), and k is a growth rate constant.
6. For the logistic growth model above, we will use 𝐵 = 108 and 𝑘 = 1.386
(accepted values for E. coli). Also, to consider the worst-case scenario, let’s
assume that our E. coli culture starts off with only one bacterium per mL of
solution (i.e. we will use 𝐴 = 1).
a. Verify that the function
𝑃=
108
1 + 99,999,999𝑒 −1.386𝑡
satisfies the logistic IVP model above. (HINT: Use this function to
compute the left-hand side of the DE, then use it (the function) to compute
the right-hand side of the DE, then confirm that the two sides are the
same. Then, verify the initial condition.)
b. According to this model, what is the long-term E. coli density in the
culture?
c. Use two steps of the Bisection Method6 (by-hand) to estimate the time at
which the E. coli culture reaches 98% of its long-term density.
d. Use algebra to compute the exact value of the time at which the E. coli
culture reaches 98% of its long-term density.
e. Would culturing a single E. coli bacterium overnight be long enough to
reach 98% of the long-term density? Does this vindicate Dr. Hill, or get
her in trouble?
The Bisection Method via Computer:
NOTE: The first thing to do for this portion of this assignment is to download and
study the Mathematica notebook Bisection_tool.nb that accompanies this homework
assignment. Then, proceed with the exercises below.
7. For the “bologna bacteria” model in Exercise 5 (with 𝑃0 = 100), use the
Mathematica tool Bisection_tool.nb to estimate the time at which the bacteria
population first reaches 90% of its long-term size. Make sure your time-estimate
is accurate to four decimal places.
Pharmacokinetics models. A number of mathematical models have been developed
to predict drug concentrations inside the body over time. This field is called
pharmacokinetics, and includes a wide-range of important questions, from the
development of therapeutic drug-treatment protocols to understanding and treating
drug addiction. For example, a 1991 paper7 reports a study of drug users from the
viewpoint of available binding sites for the drug (i.e. sites where a drug molecule can
attach and produce the intended response in the user). Letting 𝑆 represent the
proportion of drug binding sites that are active (i.e. not already occupied by another
molecule), these researchers use the model
𝑆 = 𝑒 −2.1(0.5𝑡−1+𝑒
−0.5𝑡 )
.
Here, t is the number of hours since the drug was administered.
8. For the preceding model relating S to t,
6
If you need a little more detail about the Bisection Method, read through the document
Bisection_Reading.pdf, posted with this assignment.
7
Hoppensteadt FC and Murray JD, Threshold analysis of a drug use epidemic model, Mathematical
Biosciences 53 (1/2), Feb. 1981, 79-87. (Copy posted in Blackboard as HoppensteadtMurray1981.pdf.)
a. Compute 𝑆(0), and explain what it means, in terms of binding sites.
b. Use the Mathematica tool Bisection_tool.nb to estimate the time at which
the proportion of binding sites drops to one-half of its initial value. Make
sure your time-estimate is accurate to four decimal places.