In a paper entitled "Population Waves," Bernadelli [Ber] (see also [Se]) hypothesizes a type of simplified beetle that has a natural life span of 3 years. The female of this species has a survival rate of 1/2 in the first year of life, has a survival rate of 1/3 from the second to third years, and gives birth to an average of six new females before expiring at the end of the third year. A matrix can be used to show the contribution an individual female beetle makes, in a probabilistic sense, to the female population of the species by letting ai j in the matrix A = [ai j] denote the contribution that a single female beetle of age j will make to the next year's female population of age i; that is,

a. The contribution that a female beetle makes to the population 2 years hence is determined from the entries of A2, of 3 years hence from A3, and so on. Construct A2 and A3, and try to make a general statement about the contribution of a female beetle to the population in n years' time for any positive integral value of n.
b. Use your conclusions from part (a) to describe what will occur in future years to a population of these beetles that initially consists of 6000 female beetles in each of the three age groups.
c. Construct Aˆ’1, and describe its significance regarding the population of this species.

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