Message Board Assignment (Ch8) 11) Suppose that there is an insect population consisting of juveniles (insects in their first year of life), fertile adults (insects in their second year of life), and infertile adults (insects in their third year of life or higher). Let x4 (t) represent the number of juveniles, x, (t) represent the number of fertile adults, and x3(t) represent the number of infertile adults. Suppose also that the proportion of new offspring from juveniles is a, that the propottion of new offspring from fertile adults is b, that a proportion of the juveniles sutvive to become adults the following year, that a proportion d of the fertile adults survive to become infertile adults the following year, and that a proportion e of the infertile adults survive an additional year. Then the following system of difference equations can be used to model the insect population in year t. x(t+1)|=]|c x,(t) x3(t+ 1) O d el|x;(t) The 3x3 matrix above is called a Leslie Matrix, and it is commonly used in developing age-structured models for populations in ecology. x,(t+1) a b 0] [xl(t) a) Explain in words why the terms in the @, 3, @, ,, @, 3, and @3 ; positions are each 0 it this model. b) Suppose that we know thata = 0.6, b = 0.8,c = 0.9,d = 0.7, and e = 0.8 and that the initial insect population consisted of 2,000 juveniles, 3,000 fertile adults, and 700 infertile adults. How many juveniles, fertile adults, and infertile adults will there be after 4 years? ) Notice how drastically the proportions changed in part b from the initial year to year 4, as though it were approaching some new proportions of juveniles, fertile adults, and infertile adults. It turns out that the proportion is actually approaching an eigenvector of the 3x3 system. Find the Eigenvector the system in part b is approaching, and write your answer as a vector with 100 representing the proportional number of juveniles