## Mathematics & Science
Learning Center |
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## Applications of Systems of Differential Equations## Predator-Prey ProblemsOne of the most interesting applications of systems of differential equations
is the ## The Lotka-Volterra ModelThis situation may be modeled with varying levels of complexity, but a
typical model, often called a
where: *r*(*t*) represents the size of the rabbit population in tens of thousands at time*t**f*(*t*) represents the size of the fox population in thousands at time*t**r*_{o}is the initial size of the rabbit population in tens of thousands*f*_{o}is the initial size of the fox population in thousands
## Understanding the ModelCarefully consider the relationships of the variables.
- It is
**positively influenced**by the current rabbit population size, as evidenced by the term 2*r*(*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 -0.01*r*(*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.
- 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 ModelHopefully 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 |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |