Mathematical models help us understand how human populations change over time. Two important ideas in population growth are the exponential growth model and the logistic growth model.

1. Exponential Growth Model:

   • This model shows how populations can grow when there are plenty of resources available. The formula for this model looks like this:
     • ( P(t) = P_0 e^{rt} )
     • Here’s what the symbols mean:
       • ( P(t) ) = population at a certain time ( t )
       • ( P_0 ) = starting population
       • ( r ) = growth rate
       • ( e ) = a special number used in math (called Euler's number)
   • A great example of this is the world's human population. In 1900, there were about 1.6 billion people. By 2021, that number grew to over 7.9 billion. This means the population has been growing by about 1.05% each year.

2. Logistic Growth Model:

   • This model looks at how populations grow when there are limits, like not enough food or space. The formula for this model is:
     • ( P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}} )
     • In this case:
       • ( K ) = carrying capacity, or the maximum number of people the environment can support.
   • This model suggests that as resources become limited, human populations will stop growing so fast and level off. Experts believe Earth can support between 8 to 10 billion people.

3. Applications:

   • Knowing about these models helps us make predictions about future population growth, what resources we will need, and how this will affect our environment. This information is important for making decisions about cities, food supply, and keeping our planet healthy. For example, using these models can help us handle challenges like urban growth, food security, and taking care of the environment.