The spotted owl. In the forests of Canada lives the spotted owl In order to survive, each owl 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 upa simplified) mathematical model. The model contains three stages of life: i. The hatchlings (0-1 year old) ii. The juveniles (1 - 2 years old) iii 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 lying 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 lying 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, Now 1992,pp. 91-95 la 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 year Hint: Suppose r 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 Write down your matrix on the answer sheet. moment the population of owls is known: At some 33 hatchlings, 7 juveniles (or adolescents) and 100 adults. We will call this population the initial spotted owl population 1b Determine the initial vector Xo 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 eigenvectors 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 Give an expression for A" in terms of U, D and U (see also page 328 of the book). le 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. of the spatted o (new planting, less deforestation). It is hoped that the survival rate of the hatchlings will increase to 25% 1f By using the information above, change the transition matrix and repeat lc an le. Write down what you think will happen with the population of owls. to prevent t spotted owls to die, w are going to investigate what needs to be changed using the mathematical l In previous exercises it was shown that changing the survival rate of the hatchlings is the best option for survival of the owls. You need to find this unknown survival rate. Suppose we call this value a, then jka hs, with number of juveniles in the year k + 1, number of hatchlings in the year k. J&+1 - So, a % of the hatchlings will survive. 1g Write on the answer sheet what changes are needed in the transition matrix. Deter- mine for different values of a (a 0.1, 0.2, 0.3, 0.4) the eigenvalues of the transition matrix. The population of spotted owls will survive if at least one of the absolute values the eigenvalues of the transition matrix A is larger or equal to 1. In order to see whether this is the case you can plot all eigenvalues together with the unit circle 1h In this exercise we are going to make a plot of the eigenvalues together with the unit circle. You can use the following Matlab commands. (Press shift enter to start a your command window) >A1 A; A1(2,1)-0. lamb1-eig(A1); A2-A; A2(2,1)-0.2; lamb2-eig(A2) A3 A ; A3(2,1)-0.3; lamb3-eig(A3); A4-A; A4(2,1)-0.4 lamb4 eig(A4) >t-0:0.01:2*pi; z-exp (t*i); >plot (z);hold on; plot (lamb1, 'rx'); hold on; plot (lamb2, 'gx'); hold on; plot (lamb3, 'bx); hold on; plot (lamb4, 'kx') What can you say about the plot? Write down your remarks on the answer sheet. eigenvalues of a matrix values of) the matrix. We would like to determine the survivor rate a, for which the ion of owls does not vanish. 1i Now you are going to determine the minimum value of a, such that the spotted owl will not die in the long-run. Use the following commands in Matlab: >>a-0.2; Ane FA ; Anew(2,1)-a; > while abs(eigCAnew))A1 A; A1(2,1)-0. lamb1-eig(A1); A2-A; A2(2,1)-0.2; lamb2-eig(A2) A3 A ; A3(2,1)-0.3; lamb3-eig(A3); A4-A; A4(2,1)-0.4 lamb4 eig(A4) >t-0:0.01:2*pi; z-exp (t*i); >plot (z);hold on; plot (lamb1, 'rx'); hold on; plot (lamb2, 'gx'); hold on; plot (lamb3, 'bx); hold on; plot (lamb4, 'kx') What can you say about the plot? Write down your remarks on the answer sheet. eigenvalues of a matrix values of) the matrix. We would like to determine the survivor rate a, for which the ion of owls does not vanish. 1i Now you are going to determine the minimum value of a, such that the spotted owl will not die in the long-run. Use the following commands in Matlab: >>a-0.2; Ane FA ; Anew(2,1)-a; > while abs(eigCAnew))
