Linear Algebra

TopicStable PopulationsImagine a reserve park with animals from a species that we are trying to protect.The park doesn’t have a fence and so animals cross the boundary, both fromthe inside out and from the outside in. Every year, 10% of the animals frominside of the park leave and 1% of the animals from the outside find their way in.We can ask if there is a stable level: are there populations for the park and therest of the world that will stay constant over time, with the number of animalsleaving equal to the number of animals entering?Let p n be the year n population in the park and let r n be the population inthe rest of the world.p n+1 = .90p n + .01r nr n+1 = .10p n + .99r nWe have this matrix equation.( ) (p n+1 .90 .01=r n+1 .10 .99) ()p nr nThe population will be stable if p n+1 = p n and r n+1 = r n so that the matrixequation ⃗v n+1 = T⃗v n becomes ⃗v = T⃗v. We are therefore looking for eigenvectorsfor T that are associated with the eigenvalue λ = 1. The equation ⃗0 = (λI−T)⃗v =(I − T)⃗v is(0.10 −0.01−0.10 0.01which gives the eigenspace: vectors with the restriction that p = .1r. For example,if we start with a park population p = 10, 000 animals, so that the rest of theworld has r = 100 000 animals then every year ten percent of those inside willleave the park (this is a thousand animals), and every year one percent of thosefrom the rest of the world will enter the park (also a thousand animals). It isstable, self-sustaining.Now imagine that we are trying to raise the total world population of thisspecies. For instance we can try to have the world population grow at a regularrate of 1% per year. This would make the population level stable in some sense,) (pr)=(00)