For any positive values of the input I and the rate constants k, show that system (3) has a unique equilibtium solution x1=a, x2=b, x3=c where a,b and c are all positive.
Building the Model ODEs
Apply the Balance Law to the lead flow through the blood, tissue, and bone compartments diagrammed in Figure 61 .2 to obtain a system of three linear rate equations:
(Blood) = ....
(Tissues) = .....
(Bones) = ......
The intake rate I of lead into the blood from the GI tract and the lungs is a constant or a piecewise continuous function of time.
If I is a constant, then system (3) is autonomous, because the independent variable t does not appear in any rate function. The first thing one does with an autonomous system is to look for constant solutions, i.e., equilibrium solutions. We determine them algebraically by finding values for the state variables that simultaneously make all of the rate functions zero. If I and the coefficients kji are positive constants, then system (3) has a unique equilibrium solution x1 = a, x2 = b, x3 = c with a, b, and c al] positive (Problem I 2). No matter what the initial data or the rate constants kji are, the lead levels approach that equilibrium state as t > +∞ (Problem 13).
Here is the solution to this part.
Check out Cramer's rule of determinants and linear ...
ODEs and Unique Equilibrium Levels are applied to lead levels in blood. The solution is detailed and well presented. The solution received a rating of "5" from the student who posted the question.