# An Introduction to Population Ecology - Harvesting a Population with Logistic Growth

Author(s):
Brandon M. Hale and Maeve L. McCarthy
Among the most important concerns in population ecology is the effect of harvesting a natural population. Harvesting can represent reduction of the population due to hunting or capturing individuals, which in effect removes individuals from the population (Edwards & Penney, 1999). We can incorporate a percentage harvesting term by subtracting hP from our logistic equation (Giordano et al., 2003):

If you closed your maplet window after the last set of exercises, you can start again with the button at the right. If your graphing window is still open, please close it now. Either way, from the main maplet window, choose the Logistic radio button, check the “Include Harvesting” checkbox, and click “OK”. A new window opens that includes a percentage harvesting rate, entered as a decimal fraction. For example, if 20% of the population is to be harvested each year, then = 0.2. If there is to be no harvesting, set = 0, and then the model reduces to the original logistic growth model.

Although the harvesting rate may be relatively small compared to the overall population, it can still have a large impact on the population. In combination with small intrinsic rates of increase typically seen in nature, it is easy to see why human actions (direct or indirect) have resulted in many endangered species (Edwards & Penney, 1999; Giordano et al., 2003).

### Exercises

1. For the Kentucky deer population without harvesting, P= 900,000 deer, K = 1,072,764 deer, = 0.2311 per year. Try introducing some harvesting by changing the harvesting percentage to 5% or = 0.05. What happens to the population over 10 years? Reduce the harvesting to 1%. What happens?
2. What is the smallest percentage harvesting that you can have that will cause extinction with this species? You may need to increase the stop time so that you can see the long term trend. Give your answer in percentages to one decimal place.
3. Using the harvesting percentage that you found in #2, increase the value of r. What happens? Find the smallest percentage harvesting that you can have that will cause extinction with this species. Repeat this a few times. What trend do you see? How would you explain it?
4. In 2004, 124,752 deer were hunted and reported to the KDFWR, with previous years recording similar harvests. If the KDFWR targets at most 125,000 deer to be hunted every year, can the deer population be maintained?
5. What is the maximum percentage hunting which the KDFWR can allow without the population decreasing below 500,000?

Brandon M. Hale and Maeve L. McCarthy, "An Introduction to Population Ecology - Harvesting a Population with Logistic Growth," Convergence (October 2005)

## JOMA

Journal of Online Mathematics and its Applications