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## March 7, 2021

### Inverse Circular Functions (Wrap-Up)

In class we saw a problem about how the populations of rabbits and coyotes fluctuated over a period of $$24$$ years.

Qualitatively, you should notice that both populations are roughly sinusoidal, but they're "out of phase" with each other, which is pretty typical for a predator-prey model:

• At first the rabbits are at a maximum.
• Then, as the coyotes eat the rabbits, the rabbits decrease in population and the coyotes thrive.
• At some point, there are no longer enough rabbits to feed the coyotes, and the coyotes start to die off.
• As the coyote population declines, the rabbit population once again begins to surge.
• And so the cycle repeats.

To get a better handle on what's going on, it helps to come up with equations that approximately model the populations. We can come up with an equation for the rabbits: $R=5000\cos(\tfrac{2\pi}{12}t)+10000$ And then another for the coyotes: $C=300\sin(\tfrac{2\pi}{12}t)+2000$ We can get these models by taking into account the highs and lows of the populations, where along their cycle they start, and the period of $$12$$ years.

To find the times when the rabbit population equals $$12000$$, we can set $$R=12000$$ and solve: \begin{align*} 5000\cos(\tfrac{\pi}{6}t)+10000 &= 12000\\ 5000\cos(\tfrac{\pi}{6}t) &= 2000\\ \cos(\tfrac{\pi}{6}t) &= \dfrac25\\ \end{align*} Using the inverse cosine function and the unit circle gives us two cases: $\tfrac\pi6 t \approx 1.159+2\pi n\text{, }-1.159+2\pi n$ And finally multiplying through by $$\tfrac6\pi$$: $t \approx 2.214+12n\text{, }-2.215+12n$ By plugging in $$n=0,1,2,\ldots$$, we find the solutions that fall within the bounds of the study: $t \approx 2.214, 9.786, 14.214, 21.786$

Similarly, to find times when the coyote population equals $$1800$$, we can set $$C=1800$$ and solve: \begin{align*} 300\sin(\tfrac{\pi}{6}t)+2000 &= 1800\\ 300\sin(\tfrac{\pi}{6}t) &= -200\\ \sin(\tfrac{\pi}{6}t) &= -\tfrac23\\ \end{align*} Using the inverse sine function and the unit circle gives us two cases: $\tfrac\pi6t \approx -0.729+2\pi n\text{, }-2.411+2\pi n$ And finally multiplying through by $$\tfrac6\pi$$: $t \approx -1.393+12n\text{, }-4.606+12n$ Plugging in integers for $$n$$ gives us the times the population hits $$1800$$ during the study: $t \approx 7.394, 10.606, 19.394, 22.606$ Unfortunately, $$10.606-7.394=3.212$$, so the scientists do have reason to be concerned: the population of coyotes is indeed dropping below $$1800$$ for more than $$2$$ years at a time.

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