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 for 1 adult:

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). 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)

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?)