Applications of Systems of Differential Equations

Predator-Prey Problems

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:

where:

• r(t) represents the size of the rabbit population at time t
• f(t) represents the size of the fox population at time t
• r_o is the initial size of the rabbit population
• f_o is the initial size of the fox population.

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:

• It is positively influenced by the current rabbit population size, as evidenced by the term 2r(t). This obeys the standard Malthusian model that we learned about in an earlier laboratory.
• It is negatively influenced by rabbit-fox interactions, as evidenced by the term -0.01r(t)f(t). This makes sense as we consider what this would involve--the foxes eat the rabbits during a certain proportion of their mutual encounters.

f '(t), the growth rate of the fox population, is also influenced, according to the second differential equation, by two different terms:

• It is negatively related to the current fox population size, as evidenced by the -f(t) term. This seems to contradict the Malthusian model, until we recall that the proportionality constant in the model is not purely a birth rate constant, but rather a birth-death rate constant. The fact that the constant is negative in this model simply reflects the fact that the deaths outnumber the births--at least until we consider the next term.
• Here we see a positive influence from rabbit-fox interactions. The reason why fox-rabbit interactions result in a population increase for the foxes is a little more indirect than it was for the rabbits. Clearly, a fox eating a rabbit does not lead directly to a fox giving birth, (unless some extremely unique biology is going on here.) Rather, this positive influence reflects the fact that if the fox population has a plentiful food supply it is more likely to breed--a fact that has been frequently observed in actual predator populations.

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.

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