How can I answer this problem using the equation $P(t) = P(0)e^{rt}$? Not looking for the math to be done for me, I'm just a little confused with what should be assigned to what variable.

Biologists stocked a lake with $400$ fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be $3600$.

(1) The number of fish doubled in the first year. Use this to determine the parameter $b$.

(2) How long does it take before the fish population reaches half of the carrying capacity of the lake?

  • $\begingroup$ Parameter $b$?, where is it? $\endgroup$ – Vladimir Vargas Feb 4 '15 at 1:08
  • $\begingroup$ You will need to use logistic growth, a different equation. $\endgroup$ – André Nicolas Feb 4 '15 at 1:11
  • $\begingroup$ $P(0)$ is just your initial number of fish i.e. the number of fish at time $0$. It's given to you in the information. $\endgroup$ – mattos Feb 4 '15 at 1:57

(1) I think you meant $r$ instead of $b$.

Now to determine $r$, we know that $P(0)=400$, i.e the initial amount of fish and $P(1)=2(400)=800$ from the fact that the population doubled after 1 year, that is $t=1$. Plug in this data in the given equation, take natural log on both sides and you will get a value for $r$.

(2)Half of the carrying capacity is $3600/2=1800$.

We need to find $t$ such that $P(t)=1800$.

Therefor solve $1800=400 e^{rt} $ to get a value for $t$ as needed. $r$ will be known for part (1).

  • $\begingroup$ This would be the solution for an infinite carrying capacity (please see the definition of this term to understand the exercise). $\endgroup$ – Did Jan 17 '16 at 23:51

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