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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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