In the circulatory system, red blood cells are constantly being destroyed and replaced. They carry oxygen throughout the body and they must be main- tained at a constant level. The spleen filters out and destroys a fraction of the cells daily and the bone marrow produces a number proportional to the num- ber lost on the previous day. The cll count on day is modeled as follows (see Edelstein-Keshet, 1988, pp. 27, 33): R, = number of red blood cells in circulation on day t. M, = number of red blood cells produced by marrow on day t. f = fraction of red blood cells removed by spleen, 0 < f < 1. y = production constant, y > 0. The system of difference equations satisfied by R, and M, is R, = (1 - )R, + M,, M = yfR,. (a) Express the system as a matrix equation X, = AX,. Find the cigenvalues of A and determine their sign. (b) Let the positive eigenvalue A, = 1 (R, is approximately constant); then what does this imply about y? (c) Let A1 = 1. Find A2. Then describe the behavior of R.
In the circulatory system, red blood cells are constantly being destroyed and replaced. They carry oxygen throughout the body and they must be main- tained at a constant level. The spleen filters out and destroys a fraction of the cells daily and the bone marrow produces a number proportional to the num- ber lost on the previous day. The cll count on day is modeled as follows (see Edelstein-Keshet, 1988, pp. 27, 33): R, = number of red blood cells in circulation on day t. M, = number of red blood cells produced by marrow on day t. f = fraction of red blood cells removed by spleen, 0 < f < 1. y = production constant, y > 0. The system of difference equations satisfied by R, and M, is R, = (1 - )R, + M,, M = yfR,. (a) Express the system as a matrix equation X, = AX,. Find the cigenvalues of A and determine their sign. (b) Let the positive eigenvalue A, = 1 (R, is approximately constant); then what does this imply about y? (c) Let A1 = 1. Find A2. Then describe the behavior of R.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Transcribed Image Text:In the circulatory system, red blood cells are constantly being destroyed and
replaced. They carry oxygen throughout the body and they must be main-
tained at a constant level. The spleen filters out and destroys a fraction of the
cells daily and the bone marrow produces a number proportional to the num-
ber lost on the previous day. The cell count on day is modeled as follows
(see Edelstein-Keshet, 1988, pp. 27, 33):
R, = number of red blood cells in circulation on day t.
M, = number of red blood cells produced by marrow on day t.
f = fraction of red blood cells removed by spleen, 0 < f < 1.
y = production constant, y > 0.
The system of difference equations satisfied by R, and M, is
%3D
%3D
R,1 = (1 - )R, + M,,
M,H = yfR,.
%3D
(a) Express the system as a matrix equation X41 = AX,. Find the cigenvalucs
of A and determine their sign.
(b) Let the positive eigenvalue A, = 1 (R, is approximately constant): then
what does this imply about y?
(c) Let Aj = 1. Find A2. Then describe the behavior of R.
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