Answer: 915 Construct the matrix diagram for each of the following models. 9. Dispersal

Chapter 8, Problem 9

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915 Construct the matrix diagram for each of the following models. 9. Dispersal A population of dragonflies occupies two ponds. Each day 20% of the individuals in pond A move to pond B, and 30% of the individuals in pond B move to pond A. Using At and Bt to denote the number of individuals in each pond at time t, we have At11 0.8At 1 0.3Bt Bt11 0.2At 1 0.7Bt 10. Mutation Each generation 5% of individuals carrying allele X mutate to carry allele Y. Using Xt and Yt to denote the number of individuals carrying each allele at time t, we have Xt11 0.95Xt Yt11 0.05Xt 1 Yt 11. Cancer progression Consider a population of women at risk of developing breast cancer. Each individuals health status can be in one of three states: (1) healthy, (2) earlystage disease, or (3) late-stage disease. Each year 0.2% of healthy females develop early-stage disease, with the rest remaining healthy. Likewise, 65% of women with earlystage disease recover through treatment and 35% develop late-stage disease. Finally, 10% of individuals with late-stage disease revert to early stage disease while the rest remain in late stage. Using ht, et, and lt to denote the number of individuals in each state at time t, we have ht11 0.998ht 1 0.65et et11 0.002ht 1 0.1lt lt11 0.35et 1 0.9lt 12. Yellow perch Yellow perch is a species of freshwater fish. Suppose a population of interest lives to a maximum age of 4 years. Only 5% of age 1 individuals survive to age 2, 20% of age 2 individuals survive to age 3, and 75% of age 3 individuals survive to age 4. Both age 3 and age 4 individuals reproduce, producing 100 and 150 age 1 offspring, respectively. Using ni,t for the number of individuals of age i at time t, we have n1,t11 100n3,t 1 150n4,t n2,t11 0.05n1,t n3,t11 0.2n2,t n4,t11 0.75n3,t 13. Drug diffusion A drug taken orally eventually diffuses into the bloodstream through the stomach. Let st be the amount of drug in the stomach at time t and bt the amount that has been absorbed into the bloodstream. Each time step, 50% of the drug is absorbed from the stomach into the bloodstream and 80% of the drug in the bloodstream is metabolized. We have st11 0.5st bt11 0.5st 1 0.2bt 14. Breeding systems Cooperatively breeding bird species are those in which some individuals forgo their own reproduction to help others reproduce. Therefore the population consists of three kinds of individuals: juveniles, helpers, and reproductives. Using jt, ht, and rt to denote the number of each kind at time t, a model for the population size of each is jt11 2rt ht11 0.9jt 1 0.8ht 1 0.1rt rt11 0.1jt 1 0.2ht 1 0.9rt 15. Ecological succession is a process by which a biological community progresses between different states. Suppose the community can be either primarily grassland, primarily shrub, or primarily forest, and this can change from year to year. Using tt, st, and ft to denote the probability that the community is in each of the three possible states at time t, we have tt11 0.9tt 1 0.001st st11 0.1tt 1 0.799st 1 0.001ft ft11 0.2st 1 0.999ft