For the study of a particular drug, Marcus proposes an experiment with a particular population. Approximately 200,000 people in
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Part (a)
Enter the data into your calculator or a spreadsheet. Then perform an exponential regression to find the curve of best fit. This is the closest exponential function we can assign to each population, to measure it's approximate growth.
Let functions f, g and h represent group 1 through 3 in that exact order. You should get these approximate functions:
- f(x) = 6830.52397(0.92755)^x
- g(x) = 3969.81548(1.00295)^x
- h(x) = 5.67328(1.42775)^x
Note: Group 3 doesn't have its population start until 1995, which is 10 years after 1985. This means h(x) only works when or it's meaningless otherwise.
Each function found is of the form y = a*b^x, where 'a' is the initial population (at time x = 0) and b helps determine the growth rate. We set b equal to 1+r and solve for r to get the growth rate.
For f(x), we have b = 0.92755 approximately. Set that equal to 1+r and solve for r
1+r = 0.92755
r = 0.92755-1
r = -0.07245
The negative growth rate indicates we have population decline. Specifically, the rate of decline is about 7.245%
Repeat those steps for the b value of g(x) and you should find that r = 0.00295 meaning we have roughly a 0.295% increase for Mussel group 2.
Group 3 has a growth rate of about 42.775% due to r = 0.42775 in this case. The steps will be the same as for group 1.
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Part (b)
The graph is shown below as an image attachment. I used GeoGebra to make the graph, and to also find the exponential regression equations.
We have x as the number of years since 1985, which is the starting point, and y is the population size. Something like (0,4800) means in the year 1985, the population is 4800 mussels (see group 1).
Keep in mind that the green curve for h(x) doesn't make sense for positive x values smaller than 10, due to the fact that the third group wasn't introduced until 10 years after the starting point of 1985.
Also, be sure not to think that the blue g(x) curve is linear. It looks like a flat line, but it's really a very stretched out curve. This is because the growth rate is very small.
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Part (c)
It appears that group 3 has caused group 1's population to decline, due to the red points (see part b) going downhill as we move from left to right. This drop-off occurs at around the time group 3 is introduced. Strangely, only group 1 is affected while group 2's population is still increasing. This increase is not much, but it's still a positive growth rate. So it's fairly likely that there are other confounding variables to consider, and it's not entirely group 3 that's causing the trouble.
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Part (d)
There are many ways that people impact their environment, whether it's a positive or negative impact. When regarding this topic of marine life, specifically mussels, one way that humans affect the population is what was mentioned earlier: The fact that the third group of mussels was introduced to the habitat. Often, though not always, if you introduce a new species to some environment, then the species of animals already living there may have a hard time coping with the newcomer. The new species would be referred to as "an invasive species". The invasive species is more aggressive at getting food to crowd out the older species that isn't used to the newcomer. This may explain why Mussel group 1 experiences a decline after 1995 when group 3 is introduced. This decline is fairly significant as the graph in part (b) shows. Group 2 doesn't seem to be affected at all since it's growth rate is fairly flat yet steadily increasing even if done slowly. It's possible that group 2 is simply more adapted to deal with competition better than group 1.
Another way humans impact the environment is through pollution. Air pollution may contribute to acid rainfall that would lead to the ocean. There's also chemical pollution such as flushing harmful chemicals down the drain, which also leads to the ocean. This could apply to the average consumer, or to very big industrial companies. For instance, in the news recently in Florida, there was a very big worry that a holding pond could contribute to a spike in algae growth. While algae are beneficial, there are some species that invade more than their fair share of resources and kill off anything else more or less. This is one example where pollution contributes to harmful consequences to unfortunate species who likely die off. On top of pollution, there's general habitat loss that also may explain why group 1 is experiencing a sharp decline (however it wouldn't explain why groups 2 and 3 are growing).
