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Supplementary Maple Script accompanying the manuscript:
Controlling Invasive Rodents via Synthetic and the Role of Polyandry
Manser et al., July 2019-----------------------------------------------------------------------------------------------------------------This is the supplementary Maple script that was used to calculate the analytical approximations outlined in Supplementary Text S3, which attempts to calculate the critical release thresholds mu_crit in a scenario where the population is monandrous (psi=0) and mice a release continuously (mu>0). Let us first load the necessary packages.QyQtSSV3aXRoRzYiNiNJL1ZlY3RvckNhbGN1bHVzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiQyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2xpbmFsZ0dGJSEiIg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEhRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZGLyUrZXhlY3V0YWJsZUdGN0Yy1. The Full Equation SystemLet us first define the probability of drive fertilisation (p) under monandry. We haveQyQ+SSJwRzYiLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRilJKF9zeXNsaWJHRiU2JC1GJzYkSSJzR0YlSSJTR0YlKiQtSSIrR0YoNiRGM0kiV0dGJSEiIiIiIg==Now let us define the three differential equations (Equation S6).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QyQ+SSNkU0c2Ii1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQtRic2JC1JIipHRig2JC1GMzYkSSJiR0YlSSJXR0YlLUYzNiRJInBHRiUjIiIiIiIjLUkiLUdGKDYjLUYzNiQtRic2JEkjbTFHRiUtRjM2JEkjbTJHRiUtRic2JC1GJzYkLUYzNiRGPkY4SSJTR0YlSSdTcHJpbWVHRiVGUEkjbXVHRiVGPQ==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. EquilibriaWe can now calculate the equilibrium conditions, we haveQyQ+SSZlcXVpbEc2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQ8JS9JI2RTR0YlIiIhL0koZFNwcmltZUdGJUYvL0kjZFdHRiVGLzwlSSJTR0YlSSdTcHJpbWVHRiVJIldHRiUhIiI=The eradication equilibrium is the first point where W=0. Let us store this under a different name for later use.QyQ+SS9lcmFkaWNhdGlvbl9lcUc2Ii1JKmFsbHZhbHVlc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjJkkmZXF1aWxHRiU2IyIiIkYvLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEhRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZGLyUrZXhlY3V0YWJsZUdGN0Yy3. Stability of the Eradication EquilibriumTo examine the stability criteria of the eradication equilibrium, we the Jacobian matrix of the equation system S6.QyQ+SSJKRzYiLUkpSmFjb2JpYW5HNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQ3JUkjZFdHRiVJI2RTR0YlSShkU3ByaW1lR0YlNyVJIldHRiVJIlNHRiVJJ1NwcmltZUdGJSIiIg==Which we evaluate at the eradication equilibrium.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We now calculate the Eigenvalues of the J1. QyQ+SSdzdGFiaWxHNiItSSxlaWdlbnZhbHVlc0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJJ2xpbmFsZ0c2JEYpSShfc3lzbGliR0YlNiNJI0oxR0YlIiIiTo calculate the critical release threshold, we solve the leading eigenvalue (first) for mu.QyQ+SSdtdXN0YXJHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUkmc29sdmVHRig2JCZJJ3N0YWJpbEdGJTYjIiIiSSNtdUdGJUYy4. Internal equilibrium and plausible approximationsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LlEhRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZGJSFHWe now look at the plausible approximations for the internal equilibrium as outlined in Supplementary Text S3. 4a. The lower boundary caseIn the first boundary (Equation S9), we have the following equation system.QyQ+SSlkV2R0X21pbkc2Ii1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQtSSIqR0YoNiQtRjE2JEkiYkdGJUkiV0dGJS1GMTYkLUYnNiQiIiItSSItR0YoNiNJInBHRiUjRjsiIiMtRj02Iy1GMTYkLUYnNiRJI20xR0YlLUYxNiRJI20yR0YlLUYnNiQtRjE2JEZBRjZJIlNHRiVGNiEiIg==QyQ+SSlkU2R0X21pbkc2Ii1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQtRic2JC1JIipHRig2JC1GMzYkSSJiR0YlSSJXR0YlLUYzNiRJInBHRiUjIiIiIiIjLUkiLUdGKDYjLUYzNiQtRic2JEkjbTFHRiUtRjM2JEkjbTJHRiUtRic2JC1GMzYkRj5GOEkiU0dGJUZOSSNtdUdGJSEiIg==We have the following equilibriaQyQ+SSplcXVpbF9taW5HNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkPCQvSSlkU2R0X21pbkdGJSIiIS9JKWRXZHRfbWluR0YlRi88JEkiU0dGJUkiV0dGJSEiIg==Let us now store the internal equilibrium.QyQ+STNpbnRlcm5hbF9lcXVpbF9taW5HNiItSSphbGx2YWx1ZXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyZJKmVxdWlsX21pbkdGJTYjIiIjISIiAgain, we calculate the Jacobian and evaluate it, this time at the internal equilibrium:QyQ+SSZKX21pbkc2Ii1JKUphY29iaWFuRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGKUkoX3N5c2xpYkdGJTYkNyRJKWRXZHRfbWluR0YlSSlkU2R0X21pbkdGJTckSSJXR0YlSSJTR0YlISIiQyQ+SSdKMV9taW5HNiItSSVldmFsRyUqcHJvdGVjdGVkRzYkSSZKX21pbkdGJTwkJiZJM2ludGVybmFsX2VxdWlsX21pbkdGJTYjIiIjNiMiIiImRi1GLyEiIg==The eigenvalues of the Jacobian are as follows, and the critical release thresholds are given as QyQ+SStzdGFiaWxfbWluRzYiLUksZWlnZW52YWx1ZXNHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSSdsaW5hbGdHNiRGKUkoX3N5c2xpYkdGJTYjSSdKMV9taW5HRiUhIiI=JSFHQyQ+SSttdXN0YXJfbWluRzYiLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JCZJK3N0YWJpbF9taW5HRiU2IyIiI0kjbXVHRiUhIiI=Finally, we substitute m2 for (b/2-m1)/K,. We haveQyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUklc3Vic0dGJjYkL0kjbTJHRigtSSIqRzYkRiYvSSttb2R1bGVuYW1lR0YoSS9WZWN0b3JDYWxjdWx1c0dGJTYkLUkiK0dGMTYkLUYwNiQjIiIiIiIjSSJiR0YoLUkiLUdGMTYjSSNtMUdGKCokSSJLR0YoISIiJkkrbXVzdGFyX21pbkdGKDYjRj1GPA==4a. The upper boundary caseIn the second boundary (Equation S11), we have the following equation system.QyQ+SSlkV2R0X21heEc2Ii1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQtSSIqR0YoNiQtRjE2JEkiYkdGJUkiV0dGJS1GMTYkLUYnNiQiIiItSSItR0YoNiNJInBHRiUjRjsiIiMtRj02Iy1GMTYkLUYnNiRJI20xR0YlLUYxNiRJI20yR0YlLUYnNiQtRjE2JEZBRjYtRjE2JEZBSSJTR0YlRjYhIiI=QyQ+SSlkU2R0X21heEc2Ii1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQtRic2JC1JIipHRig2JC1GMzYkSSJiR0YlSSJXR0YlLUYzNiRJInBHRiUjIiIiIiIjLUkiLUdGKDYjLUYzNiQtRic2JEkjbTFHRiUtRjM2JEkjbTJHRiUtRic2JC1GMzYkRj5GOC1GMzYkRj5JIlNHRiVGUEkjbXVHRiUhIiI=We have the following equilibriaQyQ+SSplcXVpbF9tYXhHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkPCQvSSlkU2R0X21heEdGJSIiIS9JKWRXZHRfbWF4R0YlRi88JEkiU0dGJUkiV0dGJSEiIg==QyQ+STNpbnRlcm5hbF9lcXVpbF9tYXhHNiItSSphbGx2YWx1ZXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyZJKmVxdWlsX21heEdGJTYjIiIjISIiAgain, we calculate the Jacobian and evaluate it, this time at the internal equilibrium:QyQ+SSZKX21heEc2Ii1JKUphY29iaWFuRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGKUkoX3N5c2xpYkdGJTYkNyRJKWRXZHRfbWF4R0YlSSlkU2R0X21heEdGJTckSSJXR0YlSSJTR0YlISIiQyQ+SSdKMV9tYXhHNiItSSVldmFsRyUqcHJvdGVjdGVkRzYkSSZKX21heEdGJTwkJiZJM2ludGVybmFsX2VxdWlsX21heEdGJTYjIiIjNiMiIiImRi1GLyEiIg==The eigenvalues of the Jacobian are as follows, and the critical release thresholds are given as QyQ+SStzdGFiaWxfbWF4RzYiLUksZWlnZW52YWx1ZXNHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSSdsaW5hbGdHNiRGKUkoX3N5c2xpYkdGJTYjSSdKMV9tYXhHRiUhIiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LlEhRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZGQyQ+SSttdXN0YXJfbWF4RzYiLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JCZJK3N0YWJpbF9tYXhHRiU2IyIiI0kjbXVHRiUhIiI=If we simplify and substitute m2 for (b/2-m1)/K, we haveQyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUklc3Vic0dGJjYkL0kjbTJHRigtSSIqRzYkRiYvSSttb2R1bGVuYW1lR0YoSS9WZWN0b3JDYWxjdWx1c0dGJTYkLUkiK0dGMTYkLUYwNiQjIiIiIiIjSSJiR0YoLUkiLUdGMTYjSSNtMUdGKCokSSJLR0YoISIiJkkrbXVzdGFyX21heEdGKDYjRjxGPA==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TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc4Mzc4NTcyNjA2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjKiIkIiQsKComJSJiRyIiIiwmI0YpIiIjRiklInNHIyEiIkYsRilGKSUjbTFHRi4qJHonNiVGMCUjbTJHJSNtdUdbW1tbW1tbW11bW1ttRilcW1tbXFtbW1tbW1ttIiIlRitGLiwoKiZGKEYpRi1GKUYrRjBGKUYxRi8sJEY4RisiIiEsJEYxRi9GOkY6LCZGMEYrRjFGLiwmRjBGLkYxRi5GJQ==