Only need the answer of Q8 and Q9
1. Interpret physically the form of the harvesting rate in (b) and (c). What does the parameter A stand for physically. Which model of a harvesting rate do you think is more physical? 2. Non-dimensionalize the models to obtain the forms a) x = x(1-x) -h b) jc = a(1 - a) - hac c x = x(1-x) - hatx 3. For models (a) and (b) find the fixed points and determine their stability for different values of h. Are any of them unphysical? 4. (MATLAB) plot the bifurcation diagrams for the cases (a) and (b). Label stable, unstable branches and bifurcation points. 5. Do the normal form analysis in the cases (a) and (b) and obtain analytically the results you got using MATLAB in the item above. For model (c): 6. Show that the system can have one, two, or three fixed points, depending on the values of a and h. Classify the stability of the fixed points in each case. 7. Analyze the dynamics near x = 0 and show that the bifurcation occurs when h = a. What type of bifurcation is it? 8. Show that another bifurcation occurs when h = (a + 1)2, for a < ac, where ac is to be determined. Classify the bifurcation. 9. Plot the stability diagram of the system in (a, h) parameter space. (The stability diagrams in 2- dimensional parameter space will be shown in class on Tuesday). Can hysteresis occur in any of the stability regions?
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