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7.4 Systems of Diﬀerential Equations
http://www.amazon.com/exec/obidos/ASIN/0898714540 543 At each point in time the rate (amount per second) of diﬀusion from one cell to
the other is proportional to the concentration (amount per unit volume) of the
compound in the cell giving up the compound—say the rate of diﬀusion from
cell 1 to cell 2 is α times the concentration in cell 1, and the rate of diﬀusion
from cell 2 to cell 1 is β times the concentration in cell 2. Assume α, β > 0. It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] Problem: Determine the concentration of the compound in each cell at any
given time t, and, in the long run, determine the steady-state concentrations.
Solution: If uk = uk (t) denotes the concentration of the compound in cell k at
time t, then the statements in the above assumption are translated as follows:
du1
= rate in − rate out = βu2 − αu1 ,
dt
du2
= rate in − rate out = αu1 − βu2 ,
dt D
E where u1 (0) = 1, where u2 (0) = 0. T
H In matrix notation this system is u = Au, u(0) = c, where
A= −α
α β
−β , u= u1
u2 , and IG
R c= 1
0 . Since A is the matrix of Example 7.3.3 we can use the results from Example
7.3.3 to write the solution as
u(t) = eAt c = 1
α+β β
α Y
P so that β
α −(α+β )t
u1 (t) =
+
e
α+β
α+β O
C β
α + e−(α+β )t and u2 (t) = α
−α −β
β 1
0 , α
1 − e−(α+β )t .
α+β In the long run, the concentrations in each cell stabilize in the sense that
lim u1 (t) = t→∞ β
α+β and lim u2 (t) = t→∞ α
.
α+β An innumerable variety of physical situations can be modeled by u = Au,
and the form of the solution (7.4.6) makes it clear that the eigenvalues and
eigenvectors of A are intrinsic to the underlying physical phenomenon being
investigated. We might say that the eigenvalues and eigenvectors of A act as its
genes and chromosomes...

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