Who will win?

Algebra Level 5

A stretch of desert is populated by two species of animals, roadrunners and coyotes. The populations \(r(t)\) and \(c(t)\) of roadrunners and coyotes \(t\) years from now can be modelled by

\[\begin{eqnarray} r(t+1)&=&5r(t)-4c(t) \\ c(t+1)&=&2r(t)-c(t) \end{eqnarray} \]

If there are 30 road runners and 20 coyotes initially (at time \(t=0\)), what will happen in the long run? Find \(\displaystyle \lim_{t\to\infty}\frac{r(t)}{c(t)}\).

Enter 666 if you come to the conclusion that no such limit exists.

Hint We are told that the roadrunners and coyotes live in two separate oases that are far apart. Initially there are 10 of each species in the "Oasis of Stability", while there are 20 coyotes and 10 roadrunners in the "Oasis of Fertility". Since the given equations are linear, you can study the evolution of each population separately and then add up the result. Just find out what happens in one year!


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