Modeling Population Dynamics: Exponential Equations and Stabilization

This lesson covers writing and solving systems of nonlinear equations using exponential graphs. It reviews exponential functions, growth and decay factors, and their applications in modeling real-world scenarios. The MacArthur-Wilson theory of biogeography is discussed, focusing on species immigration and extinction. The lesson models these processes using exponential equations, demonstrating how to find equilibrium points where immigration and extinction rates stabilize.

What is the primary characteristic of a pattern that can be modeled by an exponential function?

• It involves a constant addition each time period.

• It involves repeated change by the same percentage.

• It involves a linear increase over time.

• It involves a random change each time period.

How is a growth factor related to a growth rate?

• Growth factor is 1 plus the growth rate.

• Growth factor is the growth rate multiplied by 100.

• Growth factor is the growth rate minus 1.

• Growth factor is the growth rate divided by 100.

What does a decay factor of 0.88 indicate?

• The population decreases by 88% each time period.

• The population increases by 12% each time period.

• The population remains constant.

• The population decreases by 12% each time period.

According to the MacArthur-Wilson theory, when does equilibrium occur on an island?

• When the immigration rate equals the extinction rate.

• When the immigration rate is double the extinction rate.

• When the extinction rate is zero.

• When the immigration rate is zero.

What is the initial average number of species immigrating to the island each year?

• 4.2 species

• 6.1 species

• 5.0 species

• 3.5 species

What is the growth factor for the extinction rate on the island?

• 1.15

• 1.10

• 1.05

• 0.90

At what time will the immigration and extinction rates stabilize according to the model?

• 6.1 years

• 6.133 years

• 5.2 years

• 4.5 years