Include cstdlib if you want to use system(). In your population function, you're printing out the num_pop over and over again but you're not doing anything with it, which causes an endless loop. The number of population should change throughout the loop. Same goes for your other functions.
You can just delete those three functions; they won't help you. In the main function, make a for loop that creates some variable that is equal to 1, and loops until that variable is less than or equal to num_days, incrementing x each loop. Inside the loop, use num_pop and grow_per in some kind of equation and print that out.
EDIT: I mean set num_pop equal to some equation using grow_per and print out num_pop each time. When you're done with that, you just need to add data validation using while loops.
Replace that while loop at the end with a for loop like this:
for(int x = 1; x <= num_days; x++)
That's not really important, but you might as well. Your problem is that you're not changing the new_population in the loop, so new_population remains the same throughout the loop.
I need to go to sleep. You'll have to figure the rest out alone, or maybe someone else will help you.
bump - still can't seem to figure it out. I've tried re arranging where i put the calculations but i still can't figure out how to change my variable new_population inside of the loop. If anyone could explain a little further it'd be greatly appreciated. Thank you!