What patterns do you think world population might have in the future?

There are a few ways to go about this since this is a complex question but people will be influenced by their environment. 

As you see in history, there is a pattern and history repeats itself. For example: Physicist Dr. Makio Kaku has said that there are three phases for an empire, the stage where they gather themselves together and build the empire, the second in which they establish trades and expand, often by military conquest and the third, their destruction where people may scatter.

So lets take it to the future, like I said, people are influenced by the environment. Right now, we are using up our resources and depleting them out. Our cities are growing larger and expanding its borders while the center is being ruined and used up and rusting up. So in the future, people will likely be living more in the cities and expanding outwards and then scatter when it ends up destroying itself through corruption, crime, etc. Look at Detroit in the United States. Once a thriving city, now a city in debt and in ruins.

I believe the patter for people will be to be absorbed in large cities then the following generations, they will scatter but in the long future, cities may get so wide, resources depleted but technology advancing that cities and towns will combine, urban and rural areas are mixed.

As we go in the future, you must know that the governments owns and watches over cities often.  Which means that there will be technological advances but one in which governments regulate things as you now see today. People will get together, grow and advance but will eventually scatter and the cycle continues or there may not be a place to scatter to. 

This is an example of a civilization/geography pattern and is my opinion based on some facts that have happened. You might not want to do exactly this for your assignment but you can use this as a reference example.

H=   2(area/a+b)h= 2(144/a+b)h= 2(144/18+18)h= 2(144/36)h= 2(4)h=8