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Homework 7 Due Friday Apr 14 1. Using the simple predator prey model (eqns 6.1 and 6.2) with r=0.2, =0.01, q=0.4, and =0.02, draw the state space graph and zero growth isoclines for predators and victims. For a predatory-victim system that starts with 10 of individuals each, show the initial population trajectory. victim isocline (dashed): predators = r/=20 pred. isoclind (dotted): victims = q/=20 Pedators 20 20 Victims 2. In the basic SIR model for infectious disease, the system of equations were: dS/dt = -SI dI/dt = SI – kI dR/dt = kI S, I, and R are the proportions of susceptible, infected and resistant (or recovered) individuals, is the disease transmission coefficient and k is the recovery rate of infected individuals. Use the excel spreadsheet that I demonstrated in class to model chicken pox in a population of 1000 public school children. The infectious period for chicken pox is about 5 days, so k = 0.2. Studies have shown that for chicken pox is approximately 1.3. a. If you start the population with one infected child (I=0.001), how long is the epidemic likely to last? What proportion of the children is predicted to become infected during this outbreak? The predicted number of infecteds is <1 individual (I<0.001) after 40 days (and I<0.0001 after 53 d). All 1000 children are predicted to get sick during the outbreak (R=0.9999 at end, rounds to 1000 children). b. There is now a vaccine for chicken pox. Its effect is to make some fraction of the initial population resistant, prior to the outbreak of the disease. What proportion of the population must be vaccinated in order to prevent the outbreak of chicken pox? Change the initial value of S and look at the effect on I and R. When the initial S is <= 0.15 (85% vaccinated), then the proportion infected declines steadily and never gets higher than the initial 1/1000 during the outbreak. [You might have used a stricter criterion: that no additional children should be predicted to get sick. That would require that R increase by only 1 individual during the outbreak—the recovery of the initial sick child. In that case the vaccination percentage must be 94%]. c. Now repeat that analysis assuming you have a less transmissible disease, say SARS. There is approximately 0.25. What fraction of an initially susceptible population is likely to become infected during the course of an outbreak? Look at the “recovered” column to find the fraction that had been sick at some time during the outbreak: It stabilizes at 0.377. What fraction of the population would have to be vaccinated in order to prevent disease spread? Once 20% of the population are vaccinated (initial S=0.80), the infection percentage declines steadily and never increases above the initial 0.001. (About 70% have to be vaccinated to prevent ANY of the other children from getting sick. In that case R increases by only 1 individual, the initial sick child.) 3. MacArthur and Wilson’s theory of island biogeography predicts that large islands should have more species than small islands. Using their immigration and extinction graph show why that is true. Now modify that graph to show how it could be possible to find cases where there are more species on a small island than on a large one. I or E rate a. In their model, the extinction rate (dashed) is lower in large islands than small. For a constant immigration rate(dotted) to the two islands, the large island will have more species at equilibrium than the small. Species # b. Immigration rate will be lower on islands far from the mainland source of species. It is possible for a large, far island to have fewer species than a small, close, island. Species # 4. Subtidal marine communities can show regular patterns of succession just as plant communities do. A simplified dataset showing successional transition probabilities in marine invertebrates (Hill et al. 2004) has the following transition matrix. The species studied were: Hymedesmia sp. (a sponge) Hy Crisia eburnea (a bryozoan) Cr Myxilla fimbriata (a sponge) Myx Mycale lingua (a sponge) Myc Filograna implexa (a polychaete worm) Fil Next year Hy 0.85 0.11 0.02 0.00 0.02 Hy Cr Mxy Myc Fil This year Cr 0.18 0.74 0.04 0.01 0.03 Mxy 0.06 0.07 0.84 0.00 0.03 Myc 0.02 0.06 0.01 0.90 0.01 If the community starts out with these initial abundances, what is the predicted composition of the community after one year? Species Hy Cr Mxy Myc Fil Initial abundance 10 3 5 20 30 S(t+1) = A S(t) E.g.: Species Predicted abundance next year Hy Cr Mxy Myc Fil 14.24 13.27 5.92 18.33 16.24 Hy(t+1) = 10*0.85 + 3*0.18 + 5*0.06 + 20* 0.02 + 30*0.15 = 14.24. Cr(t+1) = 10*0.11 + 3*0.74 + 5*0.07 + 20*0.06 + 30*0.28 = 13.27 Do the same for the other species. Fil 0.15 0.28 0.04 0.01 0.52