The example is taken from Blume and Simon (1994)'s Mathematics for Economicsts. I am also a bit struggling with this example and I have two further questions regarding this example.

1. The intuition behind the Leslie Population model.

"Consider the organism that lives for two years. Let b1 be the birth rate of indivdiuals in their first year and b2 the birth rate for individuals in their second year. Let d1 denote the death rate of first-year individuals, so that (1 - d1) of the first-year indivdiuals survive to year two. Let xn and yn denote the number of first-year individuals and second-year indiivduals, respectively, in year n. The dynamics over time of this populationm are described by the system of diffrence equations

xn+1 = b1xn + b2yn

yn+1 = (1 - d1) x n

Can someone explain this Leslie Population model in words?


2. Consider the coupled system of difference equations generated by a Leslie model with


(1)

The next step given i nthe textbook is:


The right change of coordinates for solving this system is

X = (1/6)x + (1/3)y
Y = (-1/6)x + (2/3)y (2)

Can some one explain how the two authors come from step (1) to step (2)? Thanks in advance.