## April 4, 2012

### 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:
$K_t=(K_t_-_1 * .002)+K_t_-_1$
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:
$N_t = N_t_-_1(.1 * ((K_t_-_1-N_t_-_1)/K_t_-_1))+N_t_-_1$
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:
$N_t = N_t_-_1(.1 * ((10,000-N_t_-_1)/10,000))+N_t_-_1$

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.

#### 1 comment:

K. said...

Great job here. You can really see how the population would respond to something like a drought, etc. Nice work.