2. We will examine here a discrete dynamical system [.e., a system that evolves in time) that also happens to be a Markov chain (see section 4.9 in 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 an 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 introducing the following variables: x, amount of drug in the bloodstream after k 10-min time intervals have passed y, amount of drug in the liver after k 10-min time intervals have passed The dynamical system that this process originates is represented by the following system: where -0,1,2,3,4,5*(0)-[*]-[] or, using matrix algebra: (k+1)=A-X(k) where (k)= -]- and A 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"¹. f. For k ≥1, find the matrix 4 [its entries depend on k], using previous results. 8. It is clear that (1)=A-X(0). Show that X(2)=A-X(0), (3)=A'-X(0) X(4) AX(0) What can you say about X(k)? h. Recall that the process gets started when X(0)= To observe the system evolving with time, find (1), (2), (3) and (4). Also here: examining the vectors you just obtained (and the trend) try to guess lim X(k) i. Find X(k) = A* -X(0) as a vector with entries in terms of k. j. Use part f) and the fact that for -1<r<l, limr=0, in order to find lim k. Is some kind of stability attained as time passes? We now seek to determine the long-term behavior: find limX(k) [we can call this vector (c)]. 1. Find the vector A-X(0)= X(+1). You can see why the vector (c) is called a steady-state vector or equilibrium vector. m. Write a statement interpreting parts k) and I) in the biological context of this problem.