Problem:
The same kind of work we did in this lecture can be applied more broadly. For example, imagine a species of creature that each year either dies or gives birth to a single offspring. That offspring itself will either give birth or die in the following year, and so on. We could represent this with the following transition matrix:
infants adults dead
infants 0 0.2 0.8
adults 3.3 0.55 0.45
dead 0 0 1
This is not properly a Markov matrix, because the middle row does not add to 1. Instead, the matrix says the following: 20% of infants survive to adulthood. Adults have a 45% annual mortality rate. Adults who survive have a litter of 6 infants (3.3 = 0.55 × 6). The techniques of the lecture can still be applied to a matrix such as this. Because the questions are most easily done by multiplying the transition matrix by an initial vector many times, you may want to do them on a spreadsheet. In Excel, the command to multiply two matrices is MMULT()
a) Show for at least 10 years, how an initial population of 2 adults grows. (How do you make the 1x3 initial vector? It reflects infants, adults and deaths; that should give you the clue) Need Assignment Help?
b) Show that if adult survival drops to 45% (for example, from overharvesting), the population dies out. (Which value in the initial vector so you change to reflect the new death rate?)