6.2.5: Predator Prey Models

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ANOTHER COMMON POPULATION MODEL is that describing the coexistence of species. For example, we could consider a population of rabbits and foxes. Left to themselves, rabbits would tend to multiply, thus

\[\dfrac{d R}{d t}=a R \nonumber \]

with \(a>0 .\) In such a model the rabbit population would grow exponentially. Similarly, a population of foxes would decay without the rabbits to feed on. So, we have that

\[\dfrac{d F}{d t}=-b F\nonumber \]

for \(b>0\).

Now, if we put these populations together on a deserted island, they would interact. The more foxes, the rabbit population would decrease. However, the more rabbits, the foxes would have plenty to eat and the population would thrive. Thus, we could model the competing populations as

\[ \begin{gathered} \dfrac{d R}{d t}=a R-c F \\ \dfrac{d F}{d t}=-b F+d R \end{gathered} \label{6.55} \]

where all of the constants are positive numbers. Studying this coupled system would lead to a study of the dynamics of these populations. The nonlinear version of this system, the Lotka-Volterra model, will be discussed in the next chapter.