Understanding System Dynamics - Modeling of Environmental Sciences | GEOG 410, Lab Reports of Geography

Download Understanding System Dynamics - Modeling of Environmental Sciences | GEOG 410 and more Lab Reports Geography in PDF only on Docsity! GEOG 410 Modeling of Environmental Systems Lab 5 Understanding System Dynamics Due time: 5:00pm, Oct 15, 2008 1 Objectives (1) Understand a multiple stock systems model. (2) Understand system behaviors using pulse, step, and ramp perturbations. (3) Gain skills in using imbedded loops in Matlab 2 The Model In this exercise, we will use each of these three disturbances to examine the dynamics of the fish population in an aquatic system as shown in Figure 3.4 on page 81 of Deaton et al. book. Susceptible Fish Exposure Rate Infectiousness Sick Fish Getting in fected RecoveringResistant Fish Death Recovery T ime Fatali ty ra te InfectedFish Contact Rate Losing resistance Resistance Time Figure: Diagram for the aquatic systems with infectious disease The system has three stocks: Susceptible Fish, Sick Fish and Resistant Fish. The Susceptible Fish can get infected with the disease after contact with a Sick Fish. Assuming each fish will contact other fish c times (Contact Rate), and there are a total of p fish in the system, among which s are sick. The probability for a fish randomly contacts a sick fish is s/p. Thus the probability for a fish randomly contact a healthy fish is (1-s/p). If a fish has c contacts with other fish, the probability for not involving a sick fish for all c contacts is (1-s/p)c. Therefore, the probability for a fish contact a least once with a sick fish among the c contacts is 1-(1-s/p)c. If we have H Susceptible Fish with c contact rate, the total at risk contact is H(1-(1-s/p)c). Outflow from Susceptible Fish is Getting_Infected=Susceptible_Fish * (1-(1-Sick_Fish/(Total_Fish))c)*Infectiousness Inflow to Susceptible Fish is: Losing_Resistance_Fish 1 The Inflow to Sick Fish is the Outflow from Susceptible Fish plus perturbation sick fish. Outflow from Sick Fish includes Death and Recover. Death=Sick_Fish*fatality Recovered_Fish = Sick_Fish reaching recover time. The Inflow to Resistance Fish is the Recovered Fish and the Outflow of Resistance Fish is the number of fish reaching resistance time. Here is how we handle conveyor in Matlab. Assume the transit time is 9 days for a Sick Fish to recover. We define an array with 9 elements. On the first day, there is only Sick Fish on the first element, nothing on the rest of the element as in the following On the second day, you the surving F1 fish (i.e. F1*(1-fatality) to the second element of the array, and the newly infected fish will occupy the first element as One day nine your sick fish array will look like: One day 10 the surving F9 fish will exist Sick Fish (F9*(1-fatality)) to become Resistant Fish. The total number of sick fish in the system is F1+F2+…+F9. Similarly you can define an conveyor for Resistant Fish with the number of elements in the array equals to the Resistance Time, and move the Resistant Fish through the conveyor similarly as we move the sick fish. 3 Perturbation 3.1 Pulse Assuming we have a pulse perturbation by adding 10 Sick Fish to the systems starting on day 100 and adding another 10 Sick Fish every 10 days there after. We define an array called pulse which has as many elements as model simulation time steps. This array will have zero in the first 99 elements, and 10 on element 100, 110, 120 etc. and all the rest of the element is zero. In Matlab, we can create the pulse as in the following >>time_steps=730; >>for i=1:time_steps pulse(i)=0.0; end >> for i=100:10:time_step Pulse(i)=10; F1 0 0 0 0 0 0 0 0 F1 F2 0 0 0 0 0 0 0 F1 F2 F3 F4 F5 F6 F7 F8 F9 2 end % the end of the t-loop. Plotting the output: >>plot(time,total_sick,’-.r’, time,total_resist, ‘:b’, time, susceptible_fish, ‘-g’) % This is another way to plot multiple series of data in a single frame. Here we have specified the lines are plotted as dotted dash (-.) in red (r), dotted (:) in blue (b) and solid (-) in green (g) for total sick fish, total resistant fish and susceptible fish, respectively >>xlabel(‘Time (days)’); >>ylabel(‘Number of Fish’); >>legend(‘Sick Fish’, ‘Resistant Fish’, ‘Susceptible Fish’); % The legend has to be given in sequence as appeared in plot. The above commands were save as a .m file in data/ directory as fishpulse.m. If you copy the program to your student folder and run the model, The following figure is what you will get. 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 900 1000 Time (days) N um b er o f F is h Sick Fish Resistant Fish Susceptible Fish Exercise 1: Open the fishpulse.m program. Change the pulse pattern so that it adds 30 sick at day 100 and additional 30 sick fish every 30th day from then on. Run the program and copy the figure into your lab report and compare your figure with the figure above. Explain the differences between the two figures. 3.2 Step Perturbation 5 Exercise 2: Save the fishpulse.m program as fishstep.m. Replace the pulse array in your program with the name step. Define the step in the program such that you add 10 sick fish to the system every day starting on day 100. Run your model. You should get a figure similar to Figure 3.7 on page 82 of Deaton book. Copy your figure into your lab report and explain the model behavior you observed. 3.3 Ramp Perturbation Exercise 3: Save your fishstep.m program as fishramp.m. Replace the step array in your program with the name ramp. Define the ramp in the program such that you add 10 sick fish to the system on day 100, and 20 sick fish on day 101 and 30 fish on day 102 and so on until the end of your model simulation. Run your model. You should get a figure similar to Figure 3.9 on page 83 of Deaton book. Copy your figure into your lab report and explain the model behavior you observed. 6