2. Leslie matrices can be used to help design intervention strategies to help declining populations recover....

 

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2. Leslie matrices can be used to help design intervention strategies to help declining populations recover. One such example (Crouse et al., 1987) is on sea turtle populations that support the U.S. government-mandated use of turtle exclusion devices in shrimpers' nets. An intervention might be designed to affect any one of the entries in the Leslie matrix modeling a popula- tion. Because the dominant eigenvalue of the matrix determines the overall growth rate, we can study how changing each matrix entry affects the dominant eigenvalue, which, in turn, can describe how to formulate a policy. Consider an endangered species population of youth and adult subpopulations with the following Leslie matrix² G = (0.3 0.1 (a) Give a biological interpretation of the entries in the matrix. (b) Use either Mathematica or Matlab to find the eigenvalues and corresponding eigenvectors. Print off your Mathematica or Matlab showing the eigenvalues and eigenvectors. (c) In terms of the eigenvalues and eigenvectors, what is the general solution for a state Xt? Explain how you find the particular solution. What additional information would you need to find the particular solution? (d) Let G = (o 1.7). What values of c give a biologically meaningful model? (e) Let A₁ be the dominant eigenvalue. Find a formula for X₁ as a function of c. Plot a graph with A₁ on the y axis and c on the x axis. (f) Repeat steps (d) - (e), except replace c in the matrix above with 0.3 and 0.1 with c. (g) Repeat steps (d) - (e), except replace c in the matrix above with 0.3 and 1.7 with c. (h) Without regard to the cost of implementing any recovery plan, which entry do you think would be most effective to try to change? What biological issues might you need to understand better about the population to answer this question adequately? 2. Leslie matrices can be used to help design intervention strategies to help declining populations recover. One such example (Crouse et al., 1987) is on sea turtle populations that support the U.S. government-mandated use of turtle exclusion devices in shrimpers' nets. An intervention might be designed to affect any one of the entries in the Leslie matrix modeling a popula- tion. Because the dominant eigenvalue of the matrix determines the overall growth rate, we can study how changing each matrix entry affects the dominant eigenvalue, which, in turn, can describe how to formulate a policy. Consider an endangered species population of youth and adult subpopulations with the following Leslie matrix² G = (0.3 0.1 (a) Give a biological interpretation of the entries in the matrix. (b) Use either Mathematica or Matlab to find the eigenvalues and corresponding eigenvectors. Print off your Mathematica or Matlab showing the eigenvalues and eigenvectors. (c) In terms of the eigenvalues and eigenvectors, what is the general solution for a state Xt? Explain how you find the particular solution. What additional information would you need to find the particular solution? (d) Let G = (o 1.7). What values of c give a biologically meaningful model? (e) Let A₁ be the dominant eigenvalue. Find a formula for X₁ as a function of c. Plot a graph with A₁ on the y axis and c on the x axis. (f) Repeat steps (d) - (e), except replace c in the matrix above with 0.3 and 0.1 with c. (g) Repeat steps (d) - (e), except replace c in the matrix above with 0.3 and 1.7 with c. (h) Without regard to the cost of implementing any recovery plan, which entry do you think would be most effective to try to change? What biological issues might you need to understand better about the population to answer this question adequately?

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The entries in the matrix represent the fecundity reproductive potential of each ... View the full answer