Course:MATH110/Archive/2010-2011/003/Teams/St Gallen/Homework 12

original formula

$P(t)={\frac {1}{1+e^{-t}}}\!$

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- Change the height of the horizontal asymptote on the right and denote it by K.

In order to manipulate the height of the horizontal asymptote, you must change "K" in the denominator to a larger number if you wish for a broader asymptote and to a smaller number if you wish for a closer asymptote

Function:

$P(t)={\frac {1}{K+e^{-t}}}\!$

wider asymptote

$P(t)={\frac {1}{2+e^{-t}}}\!$

narrower asymptote

$P(t)={\frac {1}{.5+e^{-t}}}\!$

- Change the y-intercept to any number between 0 and K.

definition: Similarly to the question above, in order to manipulate the y-intercept to any other number between 0 and K you must also change the "K" in the denominator of the original formula. By inputting a number larger than 1, the y-intercept crosses on a lower intercept than in the original formula. On the other hand, if you input a smaller number in the denominator rather than 1, you may observe that the y-intercept is at a higher point on the graph.

Function:

$P(t)={\frac {1}{1+e^{-t+K}}}\!$

lower intercept

$P(t)={\frac {1}{1+e^{-t+2}}}\!$

higher intercept

$P(t)={\frac {1}{1+e^{-t+0.5}}}\!$

- BONUS - Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical.

Example Model

A small region in Antarctica holds a population of 50 polar bears, however since it's carrying capacity is 100 polar bears. Assume that the population of these polar bears are able to reproduce, by using the logistic growth model one is able to determine the exponential growth which will be obtain by the polar bear population. With the logistic growth model (displayed below) we intend to predict when the population reaches 95% of its carrying capacity.

$P(t)={\frac {50}{1+e^{-t}}}\!$

P = Population of polar bears

t = time

$100x95\%=95$ polar bears

P(t) = 95

$95={\frac {50}{1+e^{-t}}}\!$

Solution can be found in terms of t which represents the carrying capacity

• economics side note ( possibly bonus marks for this??? :) )

In actuality however, when examining the exponential growth of a particular population, let it be noted that they will never reach a specific horizontal asymptote(carrying capacity). This occurs because an individuals offspring(from that population) will offset the asymptote constantly fluctuating above and below it.

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