The spotted owl. In the forests of Canada lives the spotted owl. In order to survive, each ow needs a territory of 100 hectare. However the habitat of the spotted owl is threatened by the deforestation caused by the wood industry. In order to get some insight in the population of the spotted owl, we can set up a (simplified) mathematical model. The model contains three stages of life: i. The hatchlings (0- 1 year old) i. The juveniles (1 - 2 years old) ii. The adults (2 years and older) If the spotted owl is mature (2 years and older), they are able to lay eggs. Each year the amount of hatchlings equals 33% of the total adult population. The most important prey of the spotted owl is the so-called fying squirrel', which jumps from treetop to treetop. In normal conditions 60 % of the hatchlings will survive the dangerous first year and become a juvenile. However, due to the fast de- crease of trees in the forest there are a lot of unprotected open areas, which decreases the annual survivorship of the hatchlings. Also the number of flying squirrels' de- creases since there are less trees in the forest. The effects on the annual survivorship of the hatchlings are dramatic: a decrease form 60% to 18%. 71% of the juveniles will become an adult. The annual survivorship of an adult spotted owl is equal to 94% References: Conservation Biology, 6 (1992), pp. 505-512 and 'The great spotted owl war', Reader's Digest, Nov 1992, pp. 91-95 Read the story of the spotted owl. Let A be the transition matrix describing the change of the population for a certain year depending on the population of previous vear. Write down your matrix on the answer sheet. Hint: Suppose E R3 represents the population (state), where the first entry is the number of hatchlings, the second entry is the number of juveniles and the third entry represents the number of adults. Then Ar is the population in the nert year. At some moment the population of owls is known: 33 hatchlings, 7 juveniles (or adolescents) and 100 adults. We will call this population the initial spotted owl population. 1b Determine the initial vector x and calculate the population after 50 and 100 years. Write down your results In lab assignment 2, we already discovered how to calculate the development of the population on the long term. In the following exercises we will look at the population of spotted owls using the eigenvalues and ei of the transition matrix A By using Matlab, calculate the eigenvalues and eigenvectors of matrix A (some of the eigenvalues may be complex numbers). Use the results to rewrite the transition matrix in the form: A UDU- Hint: in Matlab, type help eig 1d Give an expression for A in terms of U, D and U-1 (see also page 328 of the book) Determine (by using Matlab) the absolute value of the eigenvalues. What can you say about the long-run population? Explain if this result depends on the initial population or not
