Mathematics & Science
Applications of Systems of Differential Equations
One of the most interesting applications of systems of differential equations is the predator-prey problem. In this laboratory we will consider an environment containing two related populations—a prey population, such as rabbits, and a predator population, such as foxes. Clearly, it is reasonable to expect that the two populations react in such a way as to influence each other's size.
The Lotka-Volterra Model
This situation may be modeled with varying levels of complexity, but a typical model, often called a Lotka-Volterra model, might look like this:
Understanding the Model
Carefully consider the relationships of the variables.
r′(t), the growth rate of the rabbit population, is influenced, according to the first differential equation, by two different terms:
f′(t), the growth rate of the fox population, is also influenced, according to the second differential equation, by two different terms:
The Accuracy of the Model
Hopefully you now have at least a little insight into the thinking that was behind the creation of the Lotka-Volterra model for predator-prey interaction. In practice, actual field studies of these types of biological systems show that the Lotka-Volterra model is a very good predictor of what actually occurs.
One early piece of research involved an analysis of the inventory of animal pelts purchased from fur trappers by the Hudson Bay Company over a lengthy period of time. Specifically, some of the pelts that were being purchased fell into a tidy predator-prey grouping, namely silver foxes, and snow-shoe hares.
If the assumption was made that the pelts being sold formed a representative sample of the total size of each population then the company's extensive records could be used to form a long term profile of the sizes of both the predator and prey populations. When this was done, and compared with the predictions made by the Lotka-Volterra model, the correlation was surprisingly good.
Well, enough theory, it's time to get to work on some problems. Let's now move on and see how we can use Mathematica to solve this kind of problem.
ODE Laboratories: A Sabbatical Project by Christopher A. Barker
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