LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKERFdG9vbHNGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=

Problem 1b plot command: Enter the desired value of A; lengthen the time interval to more than 50 if necessary (for the larger values of A) to see the long term behaviour.

(Note: Regarding the direction field in this plot: the period of the sine function is 1 year, and the time scale is 50 years or more, so the details of the field, and in some cases of the solutions, are not obvious. However, the overall trends are clear.)

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

Problem 2a: Solution commands

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

Problem 6: Phase portrait plot: Substitute the desired values of A, B, R before using. Change the initial conditions if desired.

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

The Jacobian matrix:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GUy1GLDYlUSJ4RidGL0YyLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZm4tSSVtc3VwR0YkNiVGVS1GIzYlLUkjbW5HRiQ2JFEiMkYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZYLUYsNiVRIkFGJ0YvRjItRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRlJGVEZVRmdvLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0Y9Rjk=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GUy1GLDYlUSJSRidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZSRlQtSShtZmVuY2VkR0YkNiQtRiM2LS1GLDYlUSJ5RidGL0YyLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GYW8tSSVtc3VwR0YkNiVGam4tRiM2JS1JI21uR0YkNiRRIjJGJ0Y5Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSEZgb0Ziby1GLDYlUSJCRicvRjBGPUY5RlgtRiw2JVEieEYnRi9GMkZYRmpuLyUrZXhlY3V0YWJsZUdGPUY5RjlGaXBGOQ==

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnRGZlcW0yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVElc3Vic0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYtLUYsNiVRInhGJ0YvRjItRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tSSNtbkdGJDYkUSIwRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGWi1GaG42JFEiMUYnRjlGW28tRiw2JVEjRGZGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUZccEY5

After you substitute the desired values of A, B, R into the matrix, you can use the command Eigenvalues(M), where M is the name you give to the matrix, to find the eigenvalues.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjTTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSVzdWJzRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNjItRiw2JVEiQUYnRi9GMi1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGOS1GIzYlLUZbbzYkUSIyRidGOUYvRjIvJS5saW5ldGhpY2tuZXNzR0Zdby8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Znby8lKWJldmVsbGVkR0Y9LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GYHAtRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZfcC9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJCRidGL0YyRlpGam5GYnAtRiw2JVEiUkYnRi9GMkZaLUZobjYoRmpuLUYjNiUtRltvNiRRIjhGJ0Y5Ri9GMkZjb0Zlb0Zob0Zqb0ZicC1GLDYlUSdEZmVxbTFGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUZocUY5

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEsRWlnZW52YWx1ZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSNNMUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZARj1GQA==

Problem 7: Plots and linearizations; some input is necessary. You may wish to vary the ICs for the phase portrait.

First case: b. 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 implies 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

Plot the direction field:

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

Explore the linearization about equilibrium: (You will need to substitute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImVGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImVGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRldGRA== into Dfeqm.)

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEsRWlnZW52YWx1ZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSZEZmVxbUYnRi9GMi9GM1Enbm9ybWFsRidGPS1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUrZXhlY3V0YWJsZUdGRUY9

Second case: c. 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 implies 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

Plot the phase portrait:

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

Plot an orbit which originates near the equilibrium.

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