Problem 61 - College Algebra

Question

Describe a difference between exponential growth and logistic growth.

Step-by-Step Solution

Answer

Exponential growth refers to a population that grows without limit. On the other hand, logistic growth refers to a population that grows rapidly at first, but the growth slows down and eventually levels off when the population reaches the carrying capacity - the maximum population that the environment can sustain.

Step 1: Understanding Exponential Growth

Exponential growth is a model of population growth where the size of the population increases exponentially over time. This is represented mathematically by the equation \( N(t) = N_0e^{rt} \), where \( N(t) \) is the population size at time \( t \), \( N_0 \) is the initial population size, \( r \) is the rate of growth, and \( e \) is the base of the natural logarithm.

Step 2: Understanding Logistic Growth

Logistic growth is a model of population growth where the growth rate slows down as the population approaches the carrying capacity of the environment. The mathematical representation of logistic growth is \( N(t) = \frac{K}{1 + \frac{K-N_0}{N_0} e^{-rt}} \), where \( K \) is the carrying capacity of the environment.

Step 3: Difference Between Exponential and Logistic Growth

The main difference between exponential growth and logistic growth is in how the population size changes over time. In exponential growth, the population size increases without bounds, while in logistic growth, the population size stabilizes at the carrying capacity of the environment. This is because logistic growth takes into account limitations in resources, while exponential growth does not.