![]() Now, let us think about what is happening to the hare population over time. When developing an equation, a mathematician thinks about the world. We fit the original data using mathematical methods, to estimate the values for the growth rate ( r growth), death rate ( r death), eaten rate ( r eaten), and food rate (r food), which we will use to predict the hare and lynx populations. Differential equations can be used to model the populations of lynxes (predators) and hares (prey) using the data from the Hudson Bay Company. A famous mathematician named Leonhard Euler (1707–1783) showed that differential equations could just be written as pluses and minuses, with a bit of multiplication. The equations can sometimes look very complicated, but all they are is a way to describe how and why populations change. They wanted to use mathematics to explain the rise and fall observed in the general predator-prey relationship. The predator-prey relationship was first described using differential equations by two scientists named Lotka and Volterra. ![]() Mathematicians use differential equations and data to describe what they see in the world. The question a mathematician asks is, “Can I explain this using addition, subtraction, and multiplication and can I predict the future populations” Explaining The Relationship With Mathematical Models If the predator and prey populations are balanced, they will go up and down over time, chasing each other in the circle of life. When the lynx population decreases, the hare population increases again, and the up-down cycle continues. When the hare population is low, this means less food for the lynxes and results in a decrease in the lynx population. In Figure 1, can you see that the populations of lynxes and hares fall and rise at around the same time? When there are more lynxes they eat more hares, which decreases the hare population. The rise and fall in the recorded hare and lynx populations over time suggests that there is a relationship between the two animals, which makes sense as we know that lynxes eat hares.
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