An Introduction to Population Ecology - The Explosion-Extinction Model

Author(s):
Brandon M. Hale and Maeve L. McCarthy
The logistic growth model relies on a maximum attainable level known as the carrying capacity.  There is an "opposite" of this situation, in which a population will increase only if the population remains above a particular minimal population (M). We call this model the Explosion-Extinction Model (Edwards & Penney, 1999). Mathematically, it looks similar to the Logistic Model:

The Explosion-Extinction Model is best used for species that maintain themselves by fast reproductive rates. Molles (2004) describes such species as being r-selected. These species reproduce quickly and are generally regulated by an external force, such as predation or competition with another species, rather than by internal forces, such as density dependence. Such species are able to maintain their populations as long as their predators or competitors do not reduce the population below the minimum sustainable level.

A good example of an r-selected species is the oyster mussel (Epioblasma capsaeformis), shown in Figure 6. Mollusks reproduce rapidly by releasing hundreds of larvae, which are small and free-floating in the water, leaving them vulnerable to predation. The mollusk population will be maintained as long as enough larvae survive to grow, allowing the number of adults in the population to be maintained above the minimum sustainable level.

Figure 6. Oyster mussel (Epioblasma capsaeformis)

Source: Kentucky Department of Fish and Wildlife Resources

If an r-selected species is moved to an area where their predators are not found, the result can be dramatic. An introduced species often explodes (Figure 7, left), causing the indigenous species in the area to rapidly decline to extinction (Figure 7, right) as the two species compete for the same resource.

Exercises

If your graphing window is open, close it now -- or, if the maplet is not running, you can start it again with the button at the right. Choose the Explosion-Extinction radio button in the Main window.

1. The default values are P0 = 12 units, M = 10 units, r = 0.03 per year and stop time = 100 years. Graph the population. Describe the behavior of the population over time.
2. If the initial population is reduced to 10, will the population explode, go extinct, or neither? Change the initial population to 8. What happens?
3. What do the red arrows represent? What kind of equilibrium is given by the value P = M? Describe what happens to the population when P0 > M; when P0 < M.
4. If the rate of increase is reduced to 0.01 units per year and the initial population is 7 units, when will the population become extinct? What if r is increased to 0.2 units per year?

JOMA

Journal of Online Mathematics and its Applications