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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 $k \geq 0,$ let $\mathbf{x}_{k}=\left(c_{k}, y_{k}, a_{k}\right),$ where the entries in $\mathbf{x}_{k}$ are the numbers of females in each life stage at year $k .$a. Construct the stage-matrix $A$ for the buffalo herd, such that $\mathbf{x}_{k+1}=A \mathbf{x}_{k}$ for $k \geq 0$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.

For each 100 adults, there will be approximately 38 calves and 21 yearlings.

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 6

Discrete Dynamical Systems

Vectors

Johns Hopkins University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

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