Fluctuating Migration Rate- Graph/Explanation

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.
 = Population of the previous year
 = 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.
 = The constant rate of change in the migration rate (.05)
 = **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.

Modeling population growth

 Modeling Population Growth

We have progressed during the course of this activity from simpler to increasingly complexity models. First we started of by just calculating the population of sparrow for ten years using the spreadsheet. In part two, we incorporated another variable known as "rate of increase" to refine our original model. In part three, we took it to another level by inputting a carrying capacity to create a logistic model. By observing the progress of simple to complex models, we can conclude how living systems interact, and how biological factors place limitations on something so simple to make it more complex.

Effect of Deforestation on the Sparrow Population

Due to increasing settlement on the island, deforestation has increased termendously. Settlers are effectively utilizing their surrounding resources to build new homes and a major resource is wood made by trees. However, the problem is these large beautiful trees are homes for many sparrows. Therefore, deforestation will have a significant impact on the population growth of sparrows. To investigate the growth rate, I used a spreadsheet to model the effect. First I created a new formula that will account for the deforestation effect on the population. The new formula is shown below:

So basically what this equation says is the current generation is equal to the previous generation population times the rate of increase plus the previous generation's population. Then subtracted from that is the population of the previous generation times the rate of increase plus the previous generation's population times the death rate of sparrows due to deforestation.  The death rate and the rate of increase are fixed rates, which means that these numbers do not change as the population grows. The variable R signifies the rate of increase which was equal to .01 and the variable D represent the death rate which was equal to .02. The model below illustrates the Effect of Deforestation on the the sparrow population for 20 generations.

By analyzing the graph, it is apparent that deforestation influences the sparrow growth population to decrease as the years go on.

Modeling Population Growth

The first model simply graphed the population of sparrows without any external factors. The number of individuals just grew exponentially due to a constant birth rate (10 offspring for each pair). However this model is not an accurate representation of populations in nature. Therefore, the next model introduced birth rates, death rates, and migration. By having a rate of increase (r) to account for the number of individuals entering and leaving a population, the growth model becomes more specific and accurate. Also, by adding a carrying capacity, the population model becomes even more complex and accurate. By observing nature from simple to complex, we can get a better understanding of how systems overlap or come together to work the way it does.

The first line on my graph (red) demonstrates a population of sparrows with a carrying capacity of 10,000 individuals. The second line demonstrates the population after the carrying capacity decreased to 7,000 after year 60. A probable cause for the decrease in carrying capacity is habitat destruction. Since this is a common environmental issue faces many habitats, I thought it would be interesting to see the effect on a population. The population reached its carrying capacity 10 years earlier in the second population. The formula used to determine the population after each year is:

N represents the number of individuals, t is the time in years, r is the rate of increase, and K is the carrying capacity. The rate of increase (r) accounts for births, deaths, and migration in the population and K demonstrates the maximum amount of individuals that a given environment can sustain. The graph shows an exponential growth, however after about 85 years, growth seems to slow down and remain almost constant as it approaches the carrying capacity.