A population grows according to the logistic law with a limiting population of 5×109 individuals. The initial population of 109 begins growing by doubling every hour. What will the population be after 4 hours?

Question

anonymous · Answer

109 should be $$10^9$$

anonymous · Answer

The equation I used was y' = r(1 - (y/L))y

The solution is y(t) = L / [1 + [(L/y0) - 1] ]e^(-rt)

where y0 is y naught and represents the initial population.

I initially solved for the rate using the DE. 

2*(10^9) = r(1-(10^9/5*10^9))10^9

I found r and then plugged and chugged with the solution. My answer was incorrect. Any ideas? How do I find the rate? I feel as if that is incorrect.

anonymous · Answer

Haha you're making it much more difficult than it has to be. Try just doubling the initial value 4 times?

anonymous · Answer

There is a limiting factor of 5*10^9 individuals, so that needs to be accounted for in the calculation. I have to use the equations above to calculate amount at t =4. 

I took the logistics law equation y' = r(1 - (y/L))y and used that to find the rate. (r= rate, y= population, L = limiting factor)

2*(10^9) = r(1-(10^9/5*10^9))10^9

I multiplied by 2 because the population initially doubled.

anonymous · Answer

What do you get when you double 4x?

anonymous · Answer

8*10^9 :/

anonymous · Answer

Then wouldn't it make sense that the population stops at 5e9? That's the property of the limit.

anonymous · Answer

It's carrying capacity. After a certain point, an ecosystem can't hold more individuals. At that point deaths or offspring decrease and growth stabilizes.

anonymous · Answer

Meaning a plateau, right?

anonymous · Answer

yes

anonymous · Answer

the logistics equation needs to be used

anonymous · Answer

Alright, well then by definition the highest a population can get is the limiting factor.

So, take the limit of the equation, and you'd get the 5e9, so that would be one way to use the equation to show that that is the highest it can get.