Blogs, Calculus, Modeling

Deer Season

In Pokémon Populations, we were able to provide a model for the number of Pokémon during an event based on the number of people participating.  Today, we will look at how this compares to such an event where living animals and humans are concerned.  For this example, we will look at a deer hunting season.  Then, over that season, what will the population size of deer look like?  How does this compare to our previous model?

Birth and death rates

In our post Pokémon Populations, the community event lasted for three hours.  During this time, there were many Pokémon that were born and died.  In fact the entire lifespan of these virtual entities was approximately 30 minutes.  Since the lifespan was so short compared to the time of the event, this birth and death rate had a major affect on the population of our Pokémon.

Since we are now looking at the population of deer during the course of an event, we run into a vastly different situation.  In particular, the deer firearm hunting season in Virginia was November 17 through December 1.  While this two week period is longer than the 3 hours of the community event, it is extremely short with regard to the lifespan of a deer.  Therefore, we would expect that if a deer was alive before the season, it would not likely die of natural causes during the season.  We as well may assume that there will also be no deer born during the season.  Even though it is possible such a thing would occur, the natural birth and death rates will be so overwhelmed by the effects of hunting, that we may as well ignore them for this model.

Therefore, we will work under the assumption that the population of deer would remain constant if not for hunting.  How, then, does hunting affect the population?

Hunting

When looking at how hunting will affect the population of deer, we should note that there will be a decrease in the population when a hunter successfully kills a deer.  In order to do this, the hunter must be close enough to interact with the deer.  However, such an interaction does not guarantee a successful hunt.  Therefore, what we realize is the change in population of deer is dependent upon, but not equal to, the number of interactions between deer and hunters.

Now, in order the find the number of interactions between these two, we should note that if there are more deer, there are more interactions.  Also, if there are more hunters there are more interactions.  If there were more deer but fewer hunters, or the converse, then we wouldn’t be as sure as to the number of interactions.  What we are getting at, is the the number of interactions is not just based on one population but rather we could model the number of interactions as proportional to the product of the deer population and the number of hunters.  

Cominbing these two fact we arrive at the conclusion that the change in deer population is proportional to the product of the number of hunters and the number of deer.  Therefore, if \(P\) is the deer population, \(h\) is the number of hunters and \(k\) a constant of proportionality, we get that \[\frac{dP}{dt}=kPh.\]

Number of hunters

Now, we need to look at how the number of hunters change over the hunting season.  There are a few interpretations we could have of this.  We could have that the number of hunters:

  • stays constant,
  • decreases linearly to the number of deer killed increases,
  • is proportional to the number of deer.

In this situation, each of these seems fairly reasonable.  The department of game and inland fisheries controls the number of tags given out to people in order to limit the number of deer that are harvested.  Since these tags must be purchased prior to the season, we could assume that everyone that got a tag would hunt for the duration of the season.

On the other hand, since these tags will be filled and hunters will have to stop hunting.  In this situation, we would work with the assumption that a hunter will continue to hunt until he has killed a deer.  Therefore, as deer are killed, the numbers of hunters decreases proportionally.

In the last situation, we work under the assumption that people can approximate how many deer there are.  Then the probability of getting a deer would be smaller with a smaller population of deer.  Therefore, some hunters may choose not to hunt because of the lower expected gains from the experience.  

In looking at these different scenarios, the most accurate would incorporate both the hunters being forced to stop hunting and hunters choosing to stop due to a decreased deer population.  Both cases result in the conclusion that, as deer are killed, there are fewer deer and hunters.  We may end up with a nonlinear function when combining the two, but, for the sake of simplicity in the model, we assume that we do have linear relation between the number of deer killed and the number of hunters.  That is \[\begin{align*}h&=h(P_{0})-m(\frac{dP}{dt}) \\&=h(P_{0})-m(P_{0}-P_{t}) \\ &=h(P_{0})-mP_{0}+mP(t) \\ &=mP-P_{h} ,\end{align*}\] where \(P_{0}\) is the initial deer population, \(P_{h}\) is the number deer with no hunters and \(m\) is change in hunters per change in deer.

Constants and Equations

Now that we’ve determined how the deer population will change over time, we are only left with finding our model in the given situation.  In order to do so, I will be using the numbers for populations as provided by the VDGIF

We will therefore work with a population of 1 million deer, and 300 thousand hunters at the beginning of the deer season.  In order to provide an example to work with, we will then assume that \(m=1\) and \(k=-9*10^{-8}\) with time measured in days and populations measured per deer or person.  We then get that \[\begin{align*} \frac{dP}{dt}&=-9*10^{-8}P(P-700,000) \\ \frac{dP}{P(P-700,000)}&=-9*10^{-8}dt \\ 
\int \frac{dP}{P(P-700,000)}&=\int- 9*10^{-8}dt \\ \int \frac{1}{P}+\frac{1}{700,000-P}dP&=.0625t+c \\ \ln(P)-\ln(700,000-P)&=.0625t+c \\ P&=\frac{700,000}{1+be^{-.0625t}} \\ P&=\frac{700,000}{1-\frac{3}{10}e^{-.0625t}}. \end{align*}\]  Where \(b\) is an adjusted constant based on \(c\) and found by using the initial population of the deer.  

The population of deer at end of the season would be approximately 800,000 deer.  The graph of the deer population is given below.

Comparison

In order to compare the Pokémon population over the of the community event to the deer population over hunting season, we look at both graphs.

Just a quick look at both graphs show that there is a drastic difference in the behavior of the populations.  In fact, in the game, we saw that an increase in hunters led to an increase in Pokémon, whereas the exact opposite occurred with an actual biological population.  That is, our game does not model reality.

Conclusion

Even though we had drastic differences in our model of deer population compared to the population represented in the game, both models provide for an interesting analysis in population sizes.  In using the deer population as a model in the classroom, you can provide your students with a concrete example of something that happens in nature.  On the other hand, I have always gotten my students to pay more attention by working with examples in pop culture than I have working with real examples. 

Regardless of your feeling on hunting, being able to predict the population over time is an extremely important tool.  In particular, the each state keeps track of these populations meticulously to ensure that herd of deer neither grows too large or too small.  Even though we only modeled the population over a short period, similar techniques can provide a more long term model.

I hope you enjoyed creating this model today.  If you did, be sure to like the post below, or share on Social Media.

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