## April 4, 2012

### Fluctuating Migration Rate- Graph/Explanation

$N_{t}=N_{t-1}\frac{((r_{2}-1.1)+r^i_{2})^t}{100}+N_{t-1}$

This equation represents a fluctuating migration rate to and from an island. This island loses a vast majority of its food every other year due to various, but consistent, weather conditions. The birds responded by migrating to another island for food. Every other year less birds returned to the primary island due to the lack of food. Eventually zero birds returned to the island because there was a constant source of food elsewhere.

Breaking down the equation.
$N_{t-1 }$ = Population of the previous year
$r_{2}$ = Initial rate of migration (zero)
Reasoning behind "-1.1" = This is a constant. It serves as a 110% change in population; Being negative and raising it to an exponent allows for positive changes every other year and negative changes on the off years.
$r^i_{2}$ = The constant rate of change in the migration rate (.05)
$^t$ = **The primary reason this equation prevails. By raising the numerator to the year number, immigration and emigration are introduced.(Raising a negative number to an even term makes it positive; raising it to an odd keeps it negative. This also creates a smaller number in the numerator because decimals(created by the 1.1 constant and the change of migration rate) become smaller when affected by an exponent. Creating a smaller number each time allows for an eventual emigration rate to exceed the immigration rate, bringing the population to zero.
The altered migration rate divided by 100 = The creation of a decimal prevents the migration rate from remaining at percentages over 100. Without dividing by 100 the island would hit a population size of zero after just a few years.

This activity as a whole
Throughout the three days we spent a majority of our time in class creating various graphs and data sets. The graphs progressively got more challenging to make because of the new items introduced into the equations each time. Learning how to create formulas to form graphs on population gave me the right tools I needed to construct the formula that I did. This experience was one of learning and was a big eye opener to my capabilities to extend a thought process into a visible and quantifiable set of data.