A herd of American buffalo (bison) can be modeled by a stage matrix similar to that for the spotted owls. The females can be divided into calves (up to 1 year old), yearlings (1 to 2 years), and adults. Suppose an average of 42 female calves are born each year per 100 adult females. (Only adults produce offspring.) Each year, about 60% of the calves survive, 75% of the yearlings survive, and 95% of the adults survive. For k > 0, let x = (ck, Yk, Az), where the entries in xg are the numbers of females in each life stage at year k. a. Construct the stage-matrix A for the buffalo herd, such that xg+1 = Axx for k > 0. b. [M] Show that the buffalo herd is growing, determine the expected growth rate after many years, and give the expected numbers of calves and yearlings present per 100 adults.
A herd of American buffalo (bison) can be modeled by a stage matrix similar to that for the spotted owls. The females can be divided into calves (up to 1 year old), yearlings (1 to 2 years), and adults. Suppose an average of 42 female calves are born each year per 100 adult females. (Only adults produce offspring.) Each year, about 60% of the calves survive, 75% of the yearlings survive, and 95% of the adults survive. For k > 0, let x = (ck, Yk, Az), where the entries in xg are the numbers of females in each life stage at year k. a. Construct the stage-matrix A for the buffalo herd, such that xg+1 = Axx for k > 0. b. [M] Show that the buffalo herd is growing, determine the expected growth rate after many years, and give the expected numbers of calves and yearlings present per 100 adults.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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![A herd of American buffalo (bison) can be modeled by a stage
matrix similar to that for the spotted owls. The females can be
divided into calves (up to 1 year old), yearlings (1 to 2 years),
and adults. Suppose an average of 42 female calves are
born each year per 100 adult females. (Only adults produce
offspring.) Each year, about 60% of the calves survive, 75%
of the yearlings survive, and 95% of the adults survive. For
k > 0, let x = (ck, Yk, Az), where the entries in xg are the
numbers of females in each life stage at year k.
a. Construct the stage-matrix A for the buffalo herd, such
that xg+1 = Axx for k > 0.
b. [M] Show that the buffalo herd is growing, determine
the expected growth rate after many years, and give the
expected numbers of calves and yearlings present per 100
adults.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2feae704-39b5-484c-b644-61cda9e73e92%2F370b29dc-c9cd-4282-b839-300939a1540a%2Fycryqwe_processed.png&w=3840&q=75)
Transcribed Image Text:A herd of American buffalo (bison) can be modeled by a stage
matrix similar to that for the spotted owls. The females can be
divided into calves (up to 1 year old), yearlings (1 to 2 years),
and adults. Suppose an average of 42 female calves are
born each year per 100 adult females. (Only adults produce
offspring.) Each year, about 60% of the calves survive, 75%
of the yearlings survive, and 95% of the adults survive. For
k > 0, let x = (ck, Yk, Az), where the entries in xg are the
numbers of females in each life stage at year k.
a. Construct the stage-matrix A for the buffalo herd, such
that xg+1 = Axx for k > 0.
b. [M] Show that the buffalo herd is growing, determine
the expected growth rate after many years, and give the
expected numbers of calves and yearlings present per 100
adults.
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