# Predator-Prey Population Using Sine

## Predator-Prey Modeling

Some biologists use sine and cosine curves in order to model population of a predator and its prey. On an isolated island, there are two species of mammals: lynx and hares. The lynx feeds on the hares, making them the predator and the hares the prey. Their populations adjust cyclically: the hare population decreases after being hunted, then the lynx population decreases over time due to having less food available, which in turn allows for the hares to reproduce at higher rates since they are being hunted less, causing an abundance of food for the lynx and inevitably causing an increase in their population again. In the real world, we would typically expect the two populations to taper off and find some equilibrium over time assuming that no outside factors have an effect on the two species. Below is a graphical representation of the lynx population on this island. The x-axis represents the amount of time, in weeks, with 0 corresponding to the start of observing the island. The y-axis represents the lynx population, in hundreds.

## Lynx Population

By adjusting the sliders on the graph, you can see how various outside elements can affect the changing population of the lynx on the island.
The a slider represents the amplitude of the sine wave and will adjust the maximum and minimum possible values of the population.
Changing the b slider will move the graph either up or down, representing a vertical shift of the sine wave. In terms of the island population, this corresponds to the initial population of the lynx at the start of the study.
Adjusting the k slider will either expand the graph or compress it. This k factor affects the period of the sine wave and represents the time required for the lynx population to return to its original value.

## In the Real World

Realistically, it is not common to come across a population that follows a strict sinusoidal relationship. It is more often the case that the fluctuations in population will reduce over time. The expectation is that the ecosystem will eventually reach a level of equilibrium, where the lynx population at any given time will always be "close" to a certain value.
Graphically, this can be represented using a sine wave with diminishing amplitude as opposed to the constant amplitude seen in the previous example. Such a graph can be seen below with the x and y axis still representing time in weeks and population in hundreds respectively.

In this new graph, the sliders still represent the same features:
The a slider represents the amplitude of the sine wave and will adjust the maximum and minimum possible values of the population.
Changing the b slider will move the graph either up or down, representing a vertical shift of the sine wave. In terms of the island population, this corresponds to the initial population of the lynx at the start of the study.
Adjusting the k slider will either expand the graph or compress it. This k factor affects the period of the sine wave and represents the time required for the lynx population to return to its original value.
The key difference is that we now see the heights of the waves decreasing over time and eventually the change is so small that it is almost imperceivable.

## Let's apply this knowledge

Now that you have a general idea of how sine waves can be used to represent changes in populations, try to use this new knowledge along with your previous trigonometry knowledge to complete the project found here:

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