Gopher Tortoise Population

Background

This is Part 4 of the Gopher Tortoise project. In this part, we will construct a population model for the gopher tortoise.

Data

The following table gives survival and reproduction rates for gopher tortoises at various ages. Since the real constraint on population growth is the number of females (since a relatively small number of males suffice to do their job), the data only refers to female tortoises. The survival rate is the percentage of female tortoises from that age group who survive to join the next age group. The reproduction rate is the (average) number of females born to each adult female in that age group.

Since gopher tortoises can live to be quite old (about 45 years in the wild, on average, with some living to over 60), we divide the population into 10-year age groups. Our data is based on data collected by J. Larry Landers, working in Georgia as a Project Leader for the Game Animal Research division of the International Paper Company [L]. Female gopher tortoises in his population reached sexual maturity at about 18 years of age. Their clutch sizes increased as the turtles got older and larger, and the females never went barren. We ignore tortoises 60 years old and older, since very few live that long.

Age Group	Survival Rate	Reproduction Rate
1 (0-9 years)	0.258	0
2 (10-19 years)	0.715	0.256
3 (20-29 years)	0.715	1.928
4 (30-39 years)	0.715	2.721
5 (40-49 years)	0.715	3.515
6 (50-59 years)	0.715	7.823

Problems

1. Say that g_n(t) is the number of female gopher tortoises in age group n after t time steps (decades). Using the data above, write down an expression for g₁(t) in terms of all the g_n(t-1)'s, where n ranges from 1 to 6. Do the same for the other g_n(t)'s.

2. Convert the system of linear equations you found in part 1 into a matrix equation. This will give you the transition matrix which computes the population vector in year t from the population vector in year t-1.

3. Starting with a population vector of (0, 100, 50, 50, 25, 0), trace the gopher tortoise population through a period of 100 years (10 decades). After 100 years, what is the total gopher tortoise population? What is the percentage of the population in each age group?

4. Now look at the population after 200 years, 300 years and 500 years. What is the total population in each case? In each case, what is the percentage of the population in each age group?

5. Make a table of the total gopher tortoise population for each decade from 500 years to 600 years. Using linear regression, find the exponential function which best fits this data. What is the percentage growth rate (per decade) of the population?

6. Repeat parts 3, 4 and 5 using an initial population vector of your own devising (you may have a different initial total population). How do the populations, age distributions and growth rates of the two cases compare after 500 years?

7. What conclusions can you draw about the age distribution and growth rate of the gopher tortoise population in the long term? (This, of course, assumes that the same environmental conditions persist.)

References

[L] Landers, J. Larry (Project Leader, Game Animal Research, International Paper Company), letter to Paul E. Molar, Florida Wildlife Research Lab. December 5, 1980.