(*) Fk+1 = 0.7Fk+0.1Mk and Mk+1 = -0.5Fk+ 1.3Mk where F and M are the numbers of falcons and mice in the forest, k respectively, k years after 2023.
In at tropical forest, the populations of falcons and mice are related to
each other. The mice eat everything they can while the falcons eat the mice. The population sizes of falcons and mice evolve according to the rule. * At the beginning of 2023, there were 10 falcons and 40 mice.
a) Approximately how many falcons and mice will there be at the beginning of 2068?
b)
c) Find a diagonalization, A = BDB-1, of the transition matrix A from part (b), where D is a 2x2 diagonal matrix, and B is an invertible
2x2 matrix.
d) Use your answer to (c) to estimate the (approximate) numbers
of falcons and mice there will be at the beginning of 2058.
e) what restrictions on the initial population sizes ensures the long time survival of both sources in the forest
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