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Solution: In Exercises 3–6, assume that any initial vector
Chapter 5, Problem 6E(choose chapter or problem)
Problem 6E
In Exercises 3–6, assume that any initial vector x0 has an eigenvector decomposition such that the coefficient c1 in equation (1) of this section is positive.
Show that if the predation parameter p in Exercise 5 is .5, both the owls and the squirrels will eventually perish. Find a value of p for which populations of both owls and squirrels tend toward constant levels. What are the relative population sizes in this case?
Reference:
In old-growth forests of Douglas fir, the spotted owl dines mainly on flying squirrels. Suppose the predator–prey matrix for these two populations is
Show that if the predation parameter p is .325, both populations grow. Estimate the long-term growth rate and the eventual ratio of owls to flying squirrels.
Questions & Answers
QUESTION:
Problem 6E
In Exercises 3–6, assume that any initial vector x0 has an eigenvector decomposition such that the coefficient c1 in equation (1) of this section is positive.
Show that if the predation parameter p in Exercise 5 is .5, both the owls and the squirrels will eventually perish. Find a value of p for which populations of both owls and squirrels tend toward constant levels. What are the relative population sizes in this case?
Reference:
In old-growth forests of Douglas fir, the spotted owl dines mainly on flying squirrels. Suppose the predator–prey matrix for these two populations is
Show that if the predation parameter p is .325, both populations grow. Estimate the long-term growth rate and the eventual ratio of owls to flying squirrels.
ANSWER:
Solution 6E
Step 1
Consider the predator-prey matrix for the spotted owls and flying squirrels is,
The parameter is 0.5.
The objective is to find the growth rate and the eventual ratio of owls to flying squirrels.
Find the Eigen values of the matrix :
The characteristic polynomial is,
Therefore, the Eigenvalues are 0.7 and 0.9. Here, both the Eigen values are less than the number 1 in magnitude,