4. The gas(CO,/is produced by the cellular metabolism of the body and is removed from the body when one exhales. There is a mechanism in the brain which senses the CO, level and sends signals to the breathing mechanism to take deeper breathes if the CO, level is too high. Let V, be the volume of the nth breath and let C, be the CO, concentration at the time of the nth breath. Let M be the (assumed) constant CO, concentra- tion produced by the metabolism between each breath. Then, a simple model of this system is:
4. The gas(CO,/is produced by the cellular metabolism of the body and is removed from the body when one exhales. There is a mechanism in the brain which senses the CO, level and sends signals to the breathing mechanism to take deeper breathes if the CO, level is too high. Let V, be the volume of the nth breath and let C, be the CO, concentration at the time of the nth breath. Let M be the (assumed) constant CO, concentra- tion produced by the metabolism between each breath. Then, a simple model of this system is:
Biology: The Unity and Diversity of Life (MindTap Course List)
14th Edition
ISBN:9781305073951
Author:Cecie Starr, Ralph Taggart, Christine Evers, Lisa Starr
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Chapter38: Respiration
Section: Chapter Questions
Problem 2CT
Related questions
Question
please solve (g)and (h)
![4. The gas(CO, is produced by the cellular metabolism of the body and
is removed from the body when one exhales. There is a mechanism in
the brain which senses the CO, level and sends signals to the breathing
mechanism to take deeper breathes if the CO, level is too high. Let V, be
the volume of the nth breath and let C, be the CO, concentration at the
time of the nth breath. Let M be the (assumed) constant CO2 concentra-
tion produced by the metabolism between each breath. Then, a simple
model of this system is:
Cn+1 = C, - av, + M
Va+1 = BC,
(19)
(20)
The second equation says that the volume of then+1 breath is propor-
tional to the CO, concentration at the time of the nth breath. The first
equation says that CO, concentration C, is lowered by an amount pro-
portional to the volume of the nth breath and raised by M. Given the
initial concentration Co and breath size Vo, we would like to determine
the behavior of the sequences {Cn} and {Vn}. Our discussion follows
(10], where more information can be found.
N Show that for appropriate choices of A, and A,
Ca- d, A + da
solves (23) for any choice of dj and da.
V For any choice of V, and Co, show that dị and da can be chosen so
that
C. - dịA + dzA + M/aß
V - B(d,A" + daA) + M/a
solve the original problem.
O Prove that if 4aß < 1, then C, + M/aß and V, + M/a as n +
00, so that whatever the initial concentration and breath size, the
breathing settles down to a steady rate
(g) Show that if 1 < 4aß < 4, the concentrations exhibit oscillations
which die out.
(h) Show that if a and 3 are large enough, the size of breaths and the
CO, concentration can exhibit larger and larger oscillations. Can
you explain this analytically?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6668c14f-09f9-4771-8ccf-f3d6dbec12c2%2Fb592569e-1c7d-4ef3-87e2-d4d2578188ca%2F88c50h_processed.png&w=3840&q=75)
Transcribed Image Text:4. The gas(CO, is produced by the cellular metabolism of the body and
is removed from the body when one exhales. There is a mechanism in
the brain which senses the CO, level and sends signals to the breathing
mechanism to take deeper breathes if the CO, level is too high. Let V, be
the volume of the nth breath and let C, be the CO, concentration at the
time of the nth breath. Let M be the (assumed) constant CO2 concentra-
tion produced by the metabolism between each breath. Then, a simple
model of this system is:
Cn+1 = C, - av, + M
Va+1 = BC,
(19)
(20)
The second equation says that the volume of then+1 breath is propor-
tional to the CO, concentration at the time of the nth breath. The first
equation says that CO, concentration C, is lowered by an amount pro-
portional to the volume of the nth breath and raised by M. Given the
initial concentration Co and breath size Vo, we would like to determine
the behavior of the sequences {Cn} and {Vn}. Our discussion follows
(10], where more information can be found.
N Show that for appropriate choices of A, and A,
Ca- d, A + da
solves (23) for any choice of dj and da.
V For any choice of V, and Co, show that dị and da can be chosen so
that
C. - dịA + dzA + M/aß
V - B(d,A" + daA) + M/a
solve the original problem.
O Prove that if 4aß < 1, then C, + M/aß and V, + M/a as n +
00, so that whatever the initial concentration and breath size, the
breathing settles down to a steady rate
(g) Show that if 1 < 4aß < 4, the concentrations exhibit oscillations
which die out.
(h) Show that if a and 3 are large enough, the size of breaths and the
CO, concentration can exhibit larger and larger oscillations. Can
you explain this analytically?
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