LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiQyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKERFdG9vbHNHRiUhIiI=A Tool for Finding the Jacobian Matrix of 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at the Point 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To use this tool, make copies of the commands when you want to find a Jacobian to linearize about an equilibrium, but keep this original intact for repeated use.First, define the functions 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I've used the competition model of Example 8.3.1 as an example:PkkiRkc2IiwoSSJ4R0YkIiIiKiRGJiIiIyEiIiooSSJBR0YkRidGJkYnSSJ5R0YkRidGKg==PkkiR0c2IiomSSJSR0YkIiIiLChJInlHRiRGJyokRikiIiMhIiIqKEkiQkdGJEYnSSJ4R0YkRidGKUYnRixGJw==Then define the Jacobian matrix:PkkjRGZHNiItSSdNYXRyaXhHRiQ2IzckNyQtSSVkaWZmRyUqcHJvdGVjdGVkRzYkSSJGR0YkSSJ4R0YkLUYrNiRGLkkieUdGJDckLUYrNiRJIkdHRiRGLy1GKzYkRjZGMg==Finally, evaluate the Jacobian at the equilibrium point 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LUklc3Vic0clKnByb3RlY3RlZEc2JS9JInhHNiImRic2I0kiZUdGKC9JInlHRigmRi1GKkkjRGZHRig=Now you can just substitute whatever equilibrium about which you want to linearize. For example, if we want the equilibrium at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtbkdGJDYlUSIwRicvJSVzaXplR1EjMTRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYxNiVRIjFGJ0Y0RjdGN0Y0RjdGNw==, we'd just replace 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in the previous command by 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PkkmRGZlcW1HNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYlL0kieEdGJCIiIS9JInlHRiQiIiJJI0RmR0Yk(This is just what we found in the lectures.)Finally, for specific numerical values of the parameters 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we just do another substitution step. For example, for the values used in the phase portrait on slide 4 of Example 8.3.1, we use 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(If you check the eigenvalues of this matrix, you'll see that there is one positive and one negative real, giving an unstable saddle point, as the plot would seem to indicate, although you can only see the upper half.)JSFHTTdSMApJNlJUQUJMRV9TQVZFLzQ3NzQ2MTI4MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCgiIiJGJyUieEchIiMqJiUiQUdGJyUieUdGJyEiIiwkKiglIlJHRiclIkJHRidGLEYnRi0sJComRitGJ0YoRidGLSomRjBGJywoRidGJ0YsRikqJkYxRidGKEYnRi1GJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzQ3NjA2MjE3NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCgiIiJGJyYlInhHNiMlImVHISIjKiYlIkFHRicmJSJ5R0YqRichIiIsJCooJSJSR0YnJSJCR0YnRi9GJ0YxLCQqJkYuRidGKEYnRjEqJkY0RicsKEYnRidGL0YsKiZGNUYnRihGJ0YxRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NDEzNjYwMFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCYiIiJGJyUiQUchIiIsJComJSJSR0YnJSJCR0YnRikiIiEsJEYsRilGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NDEzNjk2MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjJCIiJiEiIiQhIihGKCIiIUYoRiU=