Question

using this logistic function (as a model of population growth), how would I show that the carrying capacity will be L?

Answer 1

The function being:
\[P = \frac{ L }{ 1 + Ce ^{-kt} }\]

Answer 2

Note that for any logistic model, the population growth has a limit or approaches a certain value. This is because logistic models usually try to depict real population growth, where the growth is limited by various factors hence cannot grow infinitely but rather is limited to a certain maximum by environmental factors. Since P, the logistic model is a function of time 't', and we know that the model has a horizontal asymptote that it approaches as it goes to infinity, then the P does not grow infinitely and has a limit and this limit is the carrying capacity, i.e. the maximum population that the environment/ecosystem can support. Hence we prove that L is in fact the carrying capacity, i.e. the max population the set environment can support, as 't' approaches infinity. Evaluate this as a limit and you get:\[\bf \lim_{t \rightarrow \infty}\frac{ L }{ 1+Ce^{-kt} }=\frac{ L }{ 1+\frac{ C }{ e^{kt} } }=\frac{ L }{ 1 }=L\]Hence as the population grows over time, we see that the carrying capacity is in fact L.

@emmykim

Answer 3

thank you! that makes a lot of sense!! :) @genius12