2. We will examine here a discrete dynamical system [l.e., a system that evolves in time] that also happens to be a Markov chain (see section 4.9 is the textbook]. A drug is used to regulate liver function. A patient takes an injection containing 100 units of the drug. Every 10 minutes, 50% of the drug in the bloodstream stays in the bloodstream while 50% stays in the liver; during the same time period, 75% of the drug in the liver goes to the bloodstream while 25% stays in the liver. With in initial injection that gets the process started, 100 units of the drug go directly into the bloodstream (and 0 units into the liver). This process can be modeled mathematically by first introduding the following variables: xamount of drug in the bloodstream after k 10-min time intervals have passed y, amount of drug in the liver after k 10-nin time intervals have passed The dynamical system that this process originates is represented by the following system: where k=0,1,2,3,4,5 (0) 글 -[][] Printed at Canden Gunty College or, using matrix algebra: (k+1)-1-(k) where X(k)-[x] and and 4- a. Find all the eigenvalues and bases for all eigenspaces of the matrix A b. Why do we know that A is diagonalizable? 3414 1212 c. Find a matrix P that diagonalizes A. Why do we know that P is invertible? Then verify that PAP=D by showing that AP-PD where D is the appropriate diagonal matrix. d. Find P¹. e. Verify that A-PDP" f. For k≥1, find the matrix [its entries depend on kl, using previous results. It is clear that (I)-1-(0). Show that (2)--(0), (3)-1-(0) (4)-1-(0). What can you say about (k)? "-[C]-[] h. Recall that the process gets started when (0)- To observe the system evolving with time, ind ). X(2), (3) and (4). Also here: examining the vectors you just cotained (and the trend) try to guess lim x(k) iFind X(k)--(0) as a vector with enties in terms of k J.Use part f) and the fact that for -1<r<l, lim=0, in order to find lim k. Is some kind of stability attained as time pases? We now seek to determine the long-term behavior: find lim (k) [we can call this vector (0)]. 1. Find the vector 4-X()-(+1). You can see why the vector () is called a steady-state vector or equilibrium vector. m. Write a statement interpreting parts k) and 3 in the biological context of this problem.