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.

Population Growth Modeling

Formula:


 This formula is used to find the number of birds in a certain population.  The first part of the formula is the population of the previous generation multiplied by the rate of increase (constantly 0.1) added to the previous generation's population.  The second part is then subracted from the first part.  The second part of the formula is the population of the previous generation multiplied by the rate of increase (constantly 0.1) added to the previous generation's population and then multiplied by the rate of decrease (D) due to diseases (.15).





This simple graph was constructed using the above formula.  It shows how the population of sparrows changed over 50 years.  Even though the population of sparrows was increasing at a 0.1 rate, the rate of disease exceeded it at a rate of .15.  As time passed by, the sparrow population decreased.  As a whole, this graphing activity was a great learning experience.  As models went from simple to increasingly complex, we learned how useful spreadsheets are for keeping information organized.  It's a good way to do business from a science standpoint because  it's an efficient way to do calculations with several numbers.

Modeling Population Growth in Sparrows

This graph from year 0 to 100 show the change in population of a species of Sparrow. In the beginning of time (year 0), the population of the sparrows is at 40.  The sparrow population grows at a constant rate (r) of 0.25 with the death of 5 sparrows each generation.   At year 25, however, a natural disaster occurs and half of the Sparrow population is wiped out with the growth rate (r)  changing to 1.2. The carrying capacity (K) of the environment is 50,000 sparrows which keeps the population limited. 

Equations: 

1. Original Growth Rate Equation:

2. New Equation:

N= Population Size, t= generation, r= rate of increase, K= carrying capacity

Conclusion: Despite the great fall due to the natural disaster, the Sparrow population was able to expand greatly and seems to be leveled off due to the carrying capacity.

By using a spreadsheet to express the population of the sparrows, we're introduced to a new, easier, scientific way of graphing things. This can be used for future experiments dealing with rapid growth or exponential factors.

Sparrow Population

The formula i created shows the sparrow population fluctuating constantly; it increases and then decreases, then increases and decreases. This is because at certain points the birth/immigration rate is greater than the death/emigration rat, and vice versa. And as the years progress it continues to cycle, until it finally comes to a stop at the 99th year, and the population levels off and remains the same. The population equilibrium could be a result of many factors; the most reasonable factor could be that the birth/immigration rate became equal with the death/emigration rate. This would be an extremely ideal circumstance because this means that a decent amount of birds (2,632) could be sustained in this environment without running out of resources or exceeding the carrying capacity. 

The way we progressed during the course of this activity from simpler to increasingly complex models really made it easy to understand how to work spreadsheets very efficiently. It was a good learning process. Spreadsheets are also a good way to do business from a science standpoint because spreadsheets perform any type of calculation you type in, and everything stays organized and clear, and it processes the information quickly. It is also very easy to go back and make any corrections to your mistakes.

Population Growth Model

Equations:

Population

Increasing k

Decreasing k

This graph shows varying carrying capacities of sparrows in a particular area.  The carrying capacity increases at a constant rate for ten years and then decreases the next ten years.  The varying carrying capacity can be caused by the areas inhabitants wanting to construct new housing developments.  In their minds they think cutting trees in ten year intervals is sufficient enough to not cause any harm to the sparrows.  In actuality we can see by the graph that the sparrow population will be extinct if an event like this were to be continued.  On the graph we can see the fluctuating k-value and we see that the number of individuals fluctuates accordingly, but it is delayed at first.  The number of individuals does catch up to the lowered k-value as the years progress.  The reason behind this is that, although the carrying capacity has been lowered, the number of individuals won't tie off instantly, therefore after many years the graphs do almost become identical.  We should care because even though it may seem that the resources are being replenished, there may be some underlying problems.  We started off simple by having each pair of birds have 10 offspring and observing the results.  Next, we had a rate of increase of .1 as well as .11 on the population of the sparrows. We graphed the data and saw the difference in the two.  Next we incorporated a carrying capacity and we saw the number of individuals reach the capacity and level off.  Then I put all these things together and modeled a population in an environment with a varying carrying capacity.  Once again just like everything in biology simple things join together to become very complex.  Going from simplicity to complexity is ideal in science because it is much easier to control a smaller sample than a larger one.  Once the the sample has been perfected we can scale up.  The equations used were the population equation that takes carrying capacity into account and the two others were used to maintain a constant increase or decrease of the k-value over the respective 10 year interval.

Modeling Population Growth

     This population begins with 10 individual sparrows. From years zero to fifty, the carrying capacity for this population is two thousand. After one generation produces offspring, they die off. In year fifty, a drought occurred which limited resources and the carrying capacity decreased to one thousand. In the graph below, the two carrying capacities are shown, as well as the number of individuals within in the population throughout the years. Also the rate of increase is kept constant at .2. We can illustrate the relationship between the rate of increase, the carrying capacity and the number of individuals by this equation:

    The current population is equal to the the population of the previous generation multiplied by the rate of increase, multiplied by the carrying capacity minus the population of the previous generation divided by the carrying capacity, plus the current population.

    

      This activity has enabled us to model different scenarios on different populations. Rather than spending massive amounts of time trying to figure out the effect of certain factors when they change, with the formulas in the spreadsheets, just by changing the number, all results will change with it as well. This helps us with a visual representation of what’s going on and what is being affected, as well as saving us time. Within science, several changes occur, whether they are expected or not. Being that there are so many changes, modeling scenarios in this manner help us express the changes better and in a short period of time.

Population Growth Modeling

My graph shows the population of the sparrows for 50 generations being affected by poaching. Poaching is an illegal act of taking plants or animals for commercial use against conservatory regulations. In my graph, there is a 5 percent decrease in the sparrow population each year due to poaching. This may be a hypothetical situation, but poaching of animals can lead and have lead to extinction of species.  The rate of increase in this population is higher than the poaching rate which allows for the population to thrive despite poaching events each year. In this activity, we started with simplistic models and progressed in complexity in each step. By doing this we better understood the techniques and concepts used in the activity. As we went one step further each time, we added pieces to our last equation making it easier to do on our own and understand completely. In a scientific viewpoint this is an ideal way of doing things in business because there will be minimal error and confusion.
The two main equations used in this activity were for carrying capacity and population rate growth. K represented the carrying capacity and N was used to represent the population, t was used to show generation, r represented rate of increase each year. By changing the values for things such as rate of increase, a changing trend can be seen in the population. Depending on the different factors playing on the population, it fluctuated as seen throughout this activity. Increasing r values increased overall population and decreasing K values decreased population. The main equations are shown below.

Carrying Capacity

Population Rate Growth Model

Population Growth

The graph below compares the effect of the r value doubling half way through the experiment to r values that remain constant.  The r value is the rate of increase during a generation.  The green line on the graph shows the number of sparrows after 100 years ending at 1,315 with an r value of .05.  The pink line shows the number of sparrows after 100 years ending at 137,806 with an r value of .1. The purple lines r value doubles after 50 years, changing from .05 to .1, ending with a population of 13,462 sparrows.  The double in the r value was caused because the weather got warmer and more food was produced.  They didn't need to go and search for their food which allowed them to produce more offspring therefore leading to an increase in the rate of increase in the population.  This number is significantly lower than the population with an r value of .1 and not significantly higher than the population with an r value of .05.  I did not expect this to occur, but it is cool because something like this can really happen.  With all the weather changes that have occurred lately, this is a possibility. 

Throughout this activity the complexity of the graphs increased.  First, we modeled the population growth if each pair produced 10 offspring. Then, we looked at the change on the population when an r value was present.  Then, a new factor was added like a carrying capacity. This showed us more accurate data as to what can occur and what happens.

Formula used

This formula takes the number of individuals from the previous year and multiplies it by the rate of increase.  Then the previous years population is added to get the new total population.

Population Growth

In this Logistic model I used the same formula which was  

I started my research with 10 individuals(NT) with a Carrying Capacity  (K) of 10000 and the rate of increase(r) at .11 for 100 years. Midway I changed the Carrying Capacity (K) to 20000 and the rate of increase (r) at .1 for another 100 years. I noticed in the graph a dent was shown and the number of individuals continued to increase. This is important because the graph shows by increasing the rate and carrying capacity the number of individuals still continue to grow. Complexity models allow for you to thoroughly map trends like populations growth etc. These models are also quicker, easier and more efficient. 

Modeling Population Growth

This graph demonstrates a population growth in sparrows with the factor of a carrying capacity. The population is at a constant rate of increase but every 50 years the carrying capacity changes. The equation above was used to graph this data. To get the population you need the previous population and times it by the rate of increase and the carrying capacity at that time in the population plus the previous population. Carrying capacity is a measure of resources available to the population, since resources are limited it affects the growth of a population. As the population approaches the carrying capacity of the environment, the rate of increase of the population approaches 0. During current times this occurs in some populations of species because of the constant change in the environment the carrying capacity also changes. The lower the carrying capacity gets the lower the population will be allowed to grow, leading to a decline in a population. We should care about this because humans are a reason why a carrying capacity declines. Such as when we destroy places for species to occupy and resources for them to survive.

Modeling Population Growth

Here is the graph of my sparrow population over 300 generations (years):
What does the graph show?

At Year 0, 10 sparrows are introduced into the environment.  The sparrow population is allowed to grow at a rate of increase (r) of 0.1.  The carrying capacity (K) of the environment is originally 10,000 sparrows.  However, due to a continuous decrease in population of a competitor species, the carrying capacity for sparrows increases at a rate of .2% every year.  As the sparrow population is allowed to grow exponentially, it eventually reaches a point just below the carrying capacity.  The sparrow population and the carrying capacity then increase at the same rate until Year 200.  During Year 200, a massive wild fire wipes out a large portion of the sparrow's habitat.  This causes the carrying capacity of the environment to suddenly decrease from a high of 14,883 back down to 10,000.  As a result, the current sparrow population of 14,614 cannot be sustained, and it begins to quickly decrease to 10,000 sparrows.

This depiction of the sparrow population is important because it shows that a population of organisms cannot be maintained above the carrying capacity.  If the population becomes higher than the carrying capacity for some reason, the environment will "fix" itself, by decreasing the population until it is the same as the carrying capacity.

Equations


1.  The increase in carrying capacity of the sparrow population from Year 0 until Year 199 is shown by the equation:

K represents the carrying capacity of the environment; t represents the current year

2.  The sparrow population from Year 0 until Year 200 (when the carrying capacity is increasing at a rate of .2% every year) is shown by the equation:

N represent the population of sparrows; K represents the carrying capacity of the environment; t represents the current year

3.  The sparrow population from Year 201 until Year 300 (when the carrying capacity is at a constant 10,000 sparrows) is shown by the equation:

N represents the population of sparrows; t represents the current year


The Activity as a Whole

First, we looked at how a sparrow population can increase 5x every year without any outside factors influencing the population.  Next, we accounted for the birth rate, death rate, and immigration rate of a sparrow population.  From this information we determined the "rate of increase" of the population, which dictates how fast the population will grow.  Finally, we took into account the carrying capacity of the environment for a sparrow population.  This allowed us to see how fast the sparrow population would increase and what would happen to the population once it reached the carrying capacity.

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.

Population Growth Model

Creating models that show simulated environments is a typical way of representing projected data. By not having to actually collect real data and do field studies, mathematical models help us see what would happen in nature without us spending so many resources and so much time.

I decided to model a sparrow population with a carrying capa

city (K) starting at 10,000. Each year, due to dwindling resources, this carrying capacity decreases by .2%. At the same time, this population has a steady growth rate of .1% a year.

Here's our original population growth equation:

Our N values represent population size, R represents the rate of growth, and K represents our carrying capacity.

As many years in my model passed, eventually the population reached this diminishing carrying capacity. Our equation for this bit is here:

The population even kept going above the carrying capacity and although the population now decreased steadily alongside the carrying capacity, it was always a little higher than what the environment could handle. In year 198, however, suddenly the simulated environment changes. Now, there's a new abundance of resources that increases our carrying capacity by .5% every year. This caused the carrying capacity to shoot upward, and the population to increase along with it.

Like this:

But, in year 299, the carrying capacity regresses to its original patterns and steadily decreases at a rate of .2% a year once more.

Here's a graph of the data: