# Mathematical Biology

```Mathematical Biology
20 January 2006
Homework Problem Set #1 (KGM part)
1. Make a life cycle diagram for Garlic Mustard, an invasive weed in forests throughout much of the
Midwestern and Eastern U.S. Garlic Mustard is a biennial – after germination in early spring, the
plant forms a rosette (a group of leaves arising from a single stem at ground level), and it
overwinters in that form. In its second summer, the plant sends up a tall flowering spike, produces
several tens of flowers, sets seed, and then dies by late summer. Seeds may germinate as soon as
the next spring after they’ve formed, but they can also remain dormant for years
2. Next, make a projection matrix to represent the Garlic Mustard life cycle diagram that you
constructed above.
3. Could you use a similar kind of diagram to represent the carbon cycle of the earth? If so, draw it.
The major &quot;reservoirs&quot; for carbon (oversimplified, but ok for our purposes here) are the:
 atmosphere (as CO2)
 dissolved in water
 organisms on land (terrestrial biomass), living or dead
 organisms in water (aquatic biomass), living or dead
 carbonate rocks (formed by animals like corals and mollusks)
 coal and petroleum deposits
Carbon is incorporated into organisms via photosynthesis, and liberated from them (or their
remains) via respiration. Carbon in carbonate rocks can re-enter the atmosphere via volcanic
action. Based on the information given, include all nodes and fluxes that you think are
appropriate.
4. Consider the following life table of an organism in which we tagged 100 individuals from a
particular cohort soon after birth and followed their fates until all were dead.
Age (x, in yrs)
Number alive
0
1
2
3
4
100
50
15
2
0
Offspring
produced per
individual
0
1
20
150
0
Construct a projection matrix for this organism, based on the data given.
```