2. We will examine here a discrete dynamical system [i.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: Xx amount of drug in the bloodstream after k 10-min time intervals have passed yk 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: xk+1 Yk+1 3 Yk where k 0,1,2,3,4,5,..... = [100 0 or, using matrix algebra: X(k+1) = A-Ẵ(k) where X (k) = [] 24 and A= 1 1 2 4. h. = Recall that the process gets started when X (0) -[*]-[10]. 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 (k) i. Find (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, lim=0, in order to find lim A* 888 k. Is some kind of stability attained as time passes? We now seek to determine the long-term behavior: find lim X(k) [we can call this vector (c)]. 818 1. Find the vector A. X (∞) = X(0+1). You can see why the vector й (∞) 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.