ex2-prob3 - ( x, y, z ) = ( ky ,y , ) , Convergence of y (...

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Third problem on exam II * April 14, 2006 We assume mass action kinetics for the following chemical reaction X + Y C Y + Z, where the positive rate constants are denoted by k 1+ ,k 1 - and k 2 . Assume that X is supplied at a constant rate α > 0 and that z degrades at rate β > 0: X, Z 1. Write down the diFerential equations for the concentrations x,y,c and z . ˙ x = - k 1+ xy + k 1 - c + α ˙ y = - k 1+ xy + ( k 1 - + k 2 ) c ˙ c = k 1+ xy - ( k 1 - + k 2 ) c ˙ z = k 2 c - βz 2. Suppose that the complex c satis±es the quasi-steady-state assumption: dc dt 0 c = k 1+ k 1 - + k 2 xy Simplify the equations accordingly and show that the system describing x,y and z is linear: Plug previous expression into the x,y and z equation to get : ˙ x = - kxy + α ˙ y = 0 ˙ z = kxy - βz, where k = k 1+ k 2 k 1 - + k 2 > 0 . Since the middle equation implies that y ( t ) = y 0 , we see that upon replacing y by the constant y 0 in the remaining equations, we get a linear system. Assume that the initial concentrations of X,Y and Z are positive, and denote the initial concentration of Y by y 0 > 0. Show that the solution converges to the steady state
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Unformatted text preview: ( x, y, z ) = ( ky ,y , ) , Convergence of y ( t ) to y is trivial (recall that y ( t ) = y ), so the only thing left to show is that the x and z components converge. The remaining equations are: x =-ky x + z = ky x-z * MAP 4484 / 5489; Instructor: Patrick De Leenheer. 1 In matrix form : p x z P = p-ky ky- Pp x z P + p P By the note below, and since the matrix above has two negative eigenvalues-ky and- , we conclude that solutions of the last system converge to ( /ky ,y ,/ ). (Note: You may use the fact that if X = AX + B is a non-homogeneous linear system of arbitrary dimension n which is such that all eigenvalues of A are in the open left half plane, then all solutions converge to the steady state X =-A-1 B .) 2...
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This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.

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ex2-prob3 - ( x, y, z ) = ( ky ,y , ) , Convergence of y (...

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