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= Writing Assignment 2: Applying statistical methods to make inferences about ecological populations
#align(center, text(size: 15pt)[
  OMSB9 fall (S1) 2025 \
  Dr. Lucio Mare \
  – Aadish Verma
])

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== Part 1

=== Question 1

This ecologist uses the quadrat sampling method to estimate the population size over 15 years. Let the total area of interest be $X " m"^2$. Then, the total area sampled by the quadrats that the ecologist uses is $30 times 400 " m"^2 = 12,000 " m"^2$, and the ecologist sampled a fraction, specifically $(12,000)/X$, of the total area of interest.

At this step, the quadrat sampling method makes the key assumption that the average density of the randomly selected samples is representative of the whole population's density, enabling it to make inferences about population. this Using this assumption, we can write that the proportions of 1) sampled area vs. total area of interest and 2) total counted plants $n$ in sample vs. total population $N$ in area of interest are equal (or approximately so). We can then write #footnote([Note that we take the reciprocal, noting that neither $N$ nor $X$ can be zero for this to be valid. In practical use, this is always true, as the area of interest is nonzero and the population size is likely positive. This also applies to our calculations in Part 3.]):
$
  n/N &= (12,000" m"^2)/(X " m"^2) \
  N/n &= (X " m"^2)/(12,000" m"^2) \
  N &= n times (X " m"^2)/(12,000" m"^2).
$ <eq1>
Quadrat sampling makes more sense over alternative methods, such as mark-and-recapture, in this scenario because the shrubs being studied do not regularly move around.

=== Question 2

From 1997 through 2004, the trend of the _A. Dumosa_ population size over time perfectly matches that of logistic growth. This suggests that, around 2004, the population reached its carrying capacity of about $75,000$ individuals of _A. Dumosa_ #footnote("Although the effects of limited resources were present even before then, leading to the slow in growth that differentiates logistic growth from exponential growth."). Here is an example of a sigmoid curve that roughly matches the trend until 2005, overlaid onto the initial graph.

#image("overlaid.png")

Surprisingly, after 2005, the population begins to _decrease_ in the size; this decrease becomes more prevalent after 2008. This deviates from the "vanilla" model for logistic growth and suggests the presence of an external factor beyond just fixed resources.

=== Question 3

There are two main possibilities for what this external factor could be. The first is that it is a limiting factor that was previously present but gradually becomes more limiting over time, such as nutrient density decreasing.  The second is that this is a novel limiting factor, such as a consumer that is newly introduced into the desert. Based on this, and the knowledge that the species in question is a shrub species, we can deduce a few likely reasons for the decline in size:

1. The populations of consumers of _A. Dumosa_ increased in size, which would lead to more consumption of the shrubs and thus an increase in death rate.
2. Climate change or a similar long-term process led to a decrease in birth rates _or_ an increase in death rates of _A. Dumosa_, perhaps because the climate in some regions of the desert is no longer ideal to reproduce or grow in.
3. A disease led to an increase in death rates. This also could explain why the decline was slow at first and then increased; because shrubs are not mobile, it may have taken some time for the disease to spread to a significant amount of the population. Because disease growth is exponential, the decline would then rapidly increase after being exposed to enough "early victims."

=== Question 4

To differentiate between birth rate and death rate, the ecologist could tag the populations of _A. Dumosa_ from several randomly selected locations, and track the change. Since new _A. Dumosa_ are not tagged, only the death rate is tracked. This comes with the side effect of not tracking deaths of plants that were born after the initial sampling but died during the observation period, but we can ignore those plants as their births and deaths "cancel out" when observing the entire observation period. At the end of the observation period, we can count the number of remaining tagged plants and calculate the deaths as equal to the number of initially tagged plants minus the number of remaining tagged plants. Then, we can solve for births using $"growth" = "births" - "deaths"$ (and we already know growth from the ecologist's previous studies) and compare.

=== Question 5

All of my explanations plausibly could lead to an increase in death rate, but the second explanation is unique in that they could also lead to a decrease in birth rate, if climate change led to a decrease in fertility for the plants. If the data points to an increase in death rate rather than a decrease in birth rate, further study would be needed to properly differentiate between the two, perhaps tracking the populations of consumers and spread of disease to better pinpoint the cause.

== Part 2

Using an online calculator to calculate the variance and means of the samples, we can determine the index of dispersion, enabling us to make statements about the dispersion patterns observed #footnote("Note that, while examples for the index of dispersion often use location data, using the quadrat density frequencies is also possible as the index of disperson isn't limited to one specific data type."):


#table(
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    [*Sample*], [*Variance* #footnote("We round variance and mean to one decimal place for presentation; the unrounded numbers are used in calculations for the index of dispersion.")], [*Mean*], [*Index of dispersion*], [*Inferred pattern*]
  ),
  "1997", "13.7", "7.8", $~1.8 > 1$, "Clumped",
  "2011", "5.5", "5.3", $~1$, "Random"
)

This matches the ecologist's claim. To confirm a general trend, other researches could attempt to reproduce the ecologist's results, with different or larger samples of _A. Dumosa_; confirming the claim for such populations would strengthen the claim, as it increases the total sample size for which the trend has been confirmed. Such a change is likely due to resource availability or a similar issue; shrubs living in areas with a denser population would have significantly higher competition for resources such as nutrition, water, and sunlight. These shrubs would also have a higher death rate due to lower resources per individual; this is an example of a density-dependent population-limiting factor. Through the deaths of about 74 individuals, the population becomes more uniformly spread out, leading to lesser deaths due to density-dependent limiting factors. In effect, what we are observing is nature "selecting" for a more uniform population.

== Part 3

=== Question 1

To estimate the total number of old-growth redwood trees in Redwood National Park, given the number of trees (6,480) in a certain area of the park (320 acres out of 19,640 acres), we can use a similar method as used in #link(<eq1>, "Part 1"). Assuming the ratio of trees sampled vs. total trees is equivalent to the ratio of area sampled vs. total area, we can write and solve for the total number of trees $N$:
$
  (6,480" trees")/(N) &= (320" ac")/(19,640" ac") \
  (N)/(6,480" trees") &= (19,640" ac")/(320" ac") \
  N &= 6,480" trees" times (19,640" ac")/(320" ac") \
  &= 397,710" trees".
$

We thus estimate that 397,710 old-growth redwood trees are present in Redwood National Park.

=== Question 2

First, note that a total of $1478+416+86+10+4 = 1,994$ trees are present in the provided count. Since we are given that these are the tallest trees in the park, we know that the given data represents a portion of the right-side tail of the distribution of tree heights. Since normal distributions are symmetric there are approximately a equal number of trees at least this extreme, but on the _left_-side tail. Then, the total number of trees that are this extreme is $1,994 times 2 = 3,988$. This is approximately $(3,988)/(397,710) approx 1%$ of the total population. In other words, 99% of the population is not as extreme as the given trees; the critical value for 99% confidence is 2.576 standard deviations #footnote([Note that the critical values for this equation are based on z-scores using Figure 1, or $"DoF" approx infinity$, even though t-scores are technically the correct choice as the population standard deviation is not known. However, since the population of trees is in the thousands, t-scores are practically the same as z-scores for high degrees of freedom.]).

Similarly, a total of $2 times (416+86+10+4)/(397,710) = 0.2594855548%$ trees have a height as extreme as $330"ft"$, on either end. The corresponding confidence interval is thus for $100% - 0.2594855548% approx 99.74%$, with a critical value of 3.

From these two calculations, we know that the t-score of a tree with height $315"ft"$ is 2.576, and the t-score of a tree with height $330"ft"$ is 3. Then, their difference, or $15"ft"$, represents $3 - 2.576 = 0.424$ standard deviations. Solving:
$
  0.424 sigma = 15"ft"
  sigma = 15/0.424 "ft" approx 35.377 "ft".
$
Now, we can solve for the mean, using the fact that $330"ft"$ is $3$ standard deviations away from the mean:
$
  mu + 3 sigma &= 330"ft"
  mu &approx 223.9.
$
We conclude that the population mean $mu$ is approximately *223.9 feet*. #link("https://www.desmos.com/calculator/ngpnijdmmg", [This Desmos graph]) shows that the given data can be recovered from our derived mean and standard deviation.
=== Question 3

Yes, it is definitely plausible that the heights of the tree population is normally distributed. From an ecological standpoint, this suggests that the population mean represents a "sweet spot" for trees where several competing factors, such as necessary trunk strength, sunlight coverage, etc., balance out. However, trees still exist at the extremes as a result of biodiversity.

From a genetics standpoint, this makes sense if one considers a multitude of genes that account for tree height. If, say, there are 50 genes that increase height and 50 genes that decrease height, in a population where all 100 genes have an equal chance of being activated or deactivated, the heights of the population will approximate a normal distribution; see #link("https://www.desmos.com/calculator/19z3a0c07m", "this graph") for an example. Obviously, the genes that contribute to the height of trees are much more complex and interact with one another, but the concept holds.