Randomized discrete exponential or logistic growth models QyRJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHISIi QygtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiLUYkNiNJK1N0YXRpc3RpY3NHRihGKy1GJDYjSS5MaW5lYXJBbGdlYnJhR0YoRis= QyQ+SSVleHBvRzYiZio2I0kickdGJUYlNiNJKW9wZXJhdG9yR0YlRiVPZio2I0kiUEdGJUYlNiRGKkkmYXJyb3dHRiVGJSwmOSQiIiIqJlQkRjNGMkYzRjNGJUYlNiRGKEYyRiVGJUYlISIi QyQ+SSVsb2dpRzYiZio2JEkickdGJUkiS0dGJUYlNiNJKW9wZXJhdG9yR0YlRiVPZio2I0kiUEdGJUYlNiRGK0kmYXJyb3dHRiVGJSwmOSQiIiIqKFQkRjRGM0Y0LCZGNEY0KiZGM0Y0VCYhIiJGOkY0RjRGJUYlNiZGKEYzRik5JUYlRiVGJUY6 QyQ+SShpdGVyYXRlRzYiZio2JUkiRkdGJUkjUDBHRiVJInRHRiVGJTYjSSlyZW1lbWJlckdGJUYlQCUxOSYiIiFPOSVPLTkkNiMtRiQ2JUY1RjIsJkYvIiIiISIiRjpGJUYlRiVGOw== QyQ+SSZvcmJpdEc2ImYqNiVJIkZHRiVJI1AwR0YlSSJUR0YlNiRJImpHRiVJJG9yYkdGJUYlRiVDJT44JTcjOSU/KDgkIiIjIiIiOSZJJXRydWVHJSpwcm90ZWN0ZWRHPkYwNyQtSSNvcEdGOTYjRjAtOSQ2IyZGMDYjLCZGNEY2ISIiRjZPRjxGJUYlRiVGRQ== QyQ+SSlleHBvcmFuZEc2ImYqNiZJInJHRiVJI1AwR0YlSSJUR0YlSSJYR0YlNiVJImpHRiVJJG9yYkdGJUkjcmRHRiVGJUYlQyY+OCYtX0krU3RhdGlzdGljc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUknU2FtcGxlR0YlNiQ5JzkmPjglNyM5JT8oOCQiIiMiIiJGPEkldHJ1ZUdGNz5GPjckLUkjb3BHRjc2I0Y+LS1JJWV4cG9HRiU2IyomOSRGRCZGMjYjRkJGRDYjJkY+NiMsJkZCRkQhIiJGRE9GPjYjUVxvRXhwb25lbnRpYWx+cG9wdWxhdGlvbn5tb2RlbH5+cmFuZG9taXplZH5ieX5zdG9jaGFzdGljfnZhcmlhYmxlflhGJUYlRiVGVw== QyQ+SSlsb2dpcmFuZEc2ImYqNidJInJHRiVJIktHRiVJI1AwR0YlSSJUR0YlSSJYR0YlNiVJImpHRiVJJG9yYkdGJUkjcmRHRiVGJUYlQyY+OCYtX0krU3RhdGlzdGljc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUknU2FtcGxlR0YlNiQ5KDknPjglNyM5Jj8oOCQiIiMiIiJGPUkldHJ1ZUdGOD5GPzckLUkjb3BHRjg2I0Y/LS1JJWxvZ2lHRiU2JComOSRGRSZGMzYjRkNGRSomRlJGRTklRkU2IyZGPzYjLCZGQ0ZFISIiRkVPRj82I1FpbkxvZ2lzdGljfnBvcHVsYXRpb25+bW9kZWx+fnJhbmRvbWl6ZWR+Ynl+c3RvY2hhc3RpY352YXJpYWJsZX5YRiVGJUYlRlo= 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 Exponential growth Suppose we have counted the size of a population over T days and obtained the population sizes shown in plot1. QyY+SSNQMEc2IiIiIiEiIj5JIlRHRiUiIzlGJw== QyQ+SSJyRzYiJCIiJSEiIkYo QyQ+SSdwb3BvYnNHNiItSSlleHBvcmFuZEdGJTYmSSJyR0YlSSNQMEdGJUkiVEdGJS1JM05vcm1hbERpc3RyaWJ1dGlvbkdGJTYkIiIiJCIiJiEiIkYy QyQ+JkkiUEc2IjYjSSRtYXhHJSpwcm90ZWN0ZWRHLUklY2VpbEc2JEYpSShfc3lzbGliR0YmNiMtRig2IyZJJ3BvcG9ic0dGJjYjSSJUR0YmISIi QyQ+SSZwbG90MUc2Ii1JKnBvaW50cGxvdEdGJTYmNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSQ8LD5HRiU2JEkiakdGJSZJJ3BvcG9ic0dGJTYjRjEvRjE7IiIiSSJUR0YlL0knc3ltYm9sR0YsSSxzb2xpZGNpcmNsZUdGJS9JJmNvbG9yR0YlSSRyZWRHRiUvSStzeW1ib2xzaXplR0YlIiM6ISIi QyQtSShkaXNwbGF5RzYiNiRJJnBsb3QxR0YlL0koY2FwdGlvbkdGJVE/UG9wdWxhdGlvbn5zaXplc35kdXJpbmd+VH5kYXlzRiUiIiI= Can these data be described reasonably by an exponential growth model? We compute the growth and the per capita growth rate during each of the T-1 time intervals and try to fit it with a horizontal line (the average of the PCGRs). QyQ+SSNHUkc2Ii1JJHNlcUclKnByb3RlY3RlZEc2JCwmJkkncG9wb2JzR0YlNiMsJkkiakdGJSIiIkYwRjBGMCZGLDYjRi8hIiIvRi87RjAsJkkiVEdGJUYwRjNGMEYz QyQ+SSVQQ0dSRzYiLUkkc2VxRyUqcHJvdGVjdGVkRzYkKiYmSSNHUkdGJTYjSSJqR0YlIiIiJkkncG9wb2JzR0YlRi0hIiIvRi47Ri8sJkkiVEdGJUYvRjJGL0Yy QyQ+SSJMRzYiLUkqTGluZWFyRml0R0YlNiU3IyIiIiZJJ3BvcG9ic0dGJTYjO0YqLCZJIlRHRiVGKiEiIkYqNyNJJVBDR1JHRiVGKg== QyQ+SSJyRzYiJkkiTEdGJTYjIiIiRik= QyQtSShkaXNwbGF5RzYiNiQ3JC1JJXBsb3RHRiU2JkkickdGJS9JIlBHRiU7IiIhJkYtNiNJJG1heEclKnByb3RlY3RlZEcvSSZjb2xvckdGJUklYmx1ZUdGJS9JKnRoaWNrbmVzc0dGJSIiIy1JKnBvaW50cGxvdEdGJTYmNyMtSSRzZXFHRjM2JC1JJDwsPkdJKF9zeXNsaWJHRiU2JCZJJ3BvcG9ic0dGJTYjSSJqR0YlJkklUENHUkdGJUZHL0ZIOyIiIiwmSSJUR0YlRk0hIiJGTS9JJ3N5bWJvbEdGM0ksc29saWRjaXJjbGVHRiUvRjVJJHJlZEdGJS9JK3N5bWJvbHNpemVHRiUiIzovSShjYXB0aW9uR0YlUUZBfmhvcml6b250YWx+bGluZX5hdH50aGV+YXZlcmFnZX5QQ0dSRiVGTQ== The estimate is that the observed data can be descibed as discrete exponential growth with per capita growth rate r. We finally compare the estimated discrete exponential growth model (blue) with the actual observations (red). QyQ+SSlwb3Btb2RlbEc2Ii1JJm9yYml0R0YlNiUtSSVleHBvR0YlNiNJInJHRiUsJCZJJ3BvcG9ic0dGJTYjIiIiJCIiKSEiIkkiVEdGJUY0 QyQ+SSZwbG90Mkc2Ii1JKnBvaW50cGxvdEdGJTYmNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSQ8LD5HSShfc3lzbGliR0YlNiRJImpHRiUmSSlwb3Btb2RlbEdGJTYjRjIvRjI7IiIiSSJUR0YlL0knc3ltYm9sR0YsSSxzb2xpZGNpcmNsZUdGJS9JJmNvbG9yR0YlSSVibHVlR0YlL0krc3ltYm9sc2l6ZUdGJSIjOiEiIg== QyQtSShkaXNwbGF5RzYiNiQ3JEkmcGxvdDFHRiVJJnBsb3QyR0YlL0koY2FwdGlvbkdGJVFWT2JzZXJ2YXRpb25zfmNvbXBhcmVkfnRvfnRoZX5leHBvbmVudGlhbH5ncm93dGh+bW9kZWxGJSIiIg== We didn't estimate the P0 in the model. Try to vary P0 for a better fit. Logistic growth Suppose we have counted the size of a population over T days and obtained the population sizes shown in plot1. QyY+SSNQMEc2IiIiIiEiIj5JIlRHRiUiI0lGJw== QyY+SSJyRzYiJCIiJSEiIkYoPkkiS0dGJSIlKzdGKA== QyQ+SSdwb3BvYnNHNiItSSlsb2dpcmFuZEdGJTYnSSJyR0YlSSJLR0YlSSNQMEdGJUkiVEdGJS1JM05vcm1hbERpc3RyaWJ1dGlvbkdGJTYkIiIiJCIiJiEiIkYz QyQ+JkkiUEc2IjYjSSRtYXhHJSpwcm90ZWN0ZWRHLUklY2VpbEc2JEYpSShfc3lzbGliR0YmNiMtRig2IyZJJ3BvcG9ic0dGJjYjSSJUR0YmISIi QyQ+SSZwbG90MUc2Ii1JKnBvaW50cGxvdEdGJTYmNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSQ8LD5HSShfc3lzbGliR0YlNiRJImpHRiUmSSdwb3BvYnNHRiU2I0YyL0YyOyIiIkkiVEdGJS9JJ3N5bWJvbEdGLEksc29saWRjaXJjbGVHRiUvSSZjb2xvckdGJUkkcmVkR0YlL0krc3ltYm9sc2l6ZUdGJSIjOiEiIg== QyQtSShkaXNwbGF5RzYiNiRJJnBsb3QxR0YlL0koY2FwdGlvbkdGJVE/UG9wdWxhdGlvbn5zaXplc35kdXJpbmd+VH5kYXlzRiUiIiI= Can these data be described reasonably by an logistic growth model? We compute the growth and the per capita growth rate during each of the T-1 time intervals and try to fit it with a line (not necessarily horizontal). QyQ+SSNHUkc2Ii1JJHNlcUclKnByb3RlY3RlZEc2JCwmJkkncG9wb2JzR0YlNiMsJkkiakdGJSIiIkYwRjBGMCZGLDYjRi8hIiIvRi87RjAsJkkiVEdGJUYwRjNGMEYz QyQ+SSVQQ0dSRzYiLUkkc2VxRyUqcHJvdGVjdGVkRzYkKiYmSSNHUkdGJTYjSSJqR0YlIiIiJkkncG9wb2JzR0YlRi0hIiIvRi47Ri8sJkkiVEdGJUYvRjJGL0Yy QyQ+SSJMRzYiLUkqTGluZWFyRml0R0YlNiY3JCIiIkkiUEdGJSZJJ3BvcG9ic0dGJTYjO0YqLCZJIlRHRiVGKiEiIkYqNyNJJVBDR1JHRiVGK0Yq QyQ+SSJMRzYiLUkodW5hcHBseUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkRiRJIlBHRiUhIiI= QyQ+SSJLRzYiLUkmc29sdmVHRiU2JC8tSSJMR0YlNiNJIlBHRiUiIiFGLSIiIg== QyQ+SSJyRzYiLUkiTEdGJTYjIiIhIiIi QyQtSShkaXNwbGF5RzYiNiQ3JC1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JS1JIkxHRiU2I0kiUEdGJS9GMTsiIiEtSSRtYXhHRis2JCZGMTYjRjZJIktHRiUvSSZjb2xvckdGJUklYmx1ZUdGJS1JKnBvaW50cGxvdEdGJTYmNyMtSSRzZXFHRis2JC1JJDwsPkdGLDYkJkkncG9wb2JzR0YlNiNJImpHRiUmSSVQQ0dSR0YlRkovRks7IiIiLCZJIlRHRiVGUCEiIkZQL0knc3ltYm9sR0YrSSxzb2xpZGNpcmNsZUdGJS9GPEkkcmVkR0YlL0krc3ltYm9sc2l6ZUdGJSIjOi9JKGNhcHRpb25HRiVRSFZhbHVlc35mb3J+dGhlflBDR1JzfmZpdHRlZH53aXRofmF+bGluZUYlRlA= The estimate is that the observed data can be descibed as discrete logistic growth with per capita growth rate r and carrying capacity K. We finally compare the logistic growth model with parameters r and K (blue) with the actual observations (red). QyQ+SSlwb3Btb2RlbEc2Ii1JJm9yYml0R0YlNiUtSSVsb2dpR0YlNiRJInJHRiVJIktHRiUsJCZJJ3BvcG9ic0dGJTYjIiIiJCIiKCEiIkkiVEdGJUY1 QyQ+SSZwbG90Mkc2Ii1JKnBvaW50cGxvdEdGJTYmNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSQ8LD5HSShfc3lzbGliR0YlNiRJImpHRiUmSSlwb3Btb2RlbEdGJTYjRjIvRjI7IiIiSSJUR0YlL0knc3ltYm9sR0YsSSxzb2xpZGNpcmNsZUdGJS9JJmNvbG9yR0YlSSVibHVlR0YlL0krc3ltYm9sc2l6ZUdGJSIjOiEiIg== QyQtSShkaXNwbGF5RzYiNiQ3JEkmcGxvdDFHRiVJJnBsb3QyR0YlL0koY2FwdGlvbkdGJVFTT2JzZXJ2YXRpb25zfmNvbXBhcmVkfnRvfnRoZX5sb2dpc3RpY35ncm93dGh+bW9kZWxGJSIiIg== The choice of P0 in the model can be important for the fit of these two graphs. We didn't estimate the initial population. Try to modify P0 in the model by varying the factor in front pop_obs[1] in the model. JSFH Stochastic single population models Exponential growth QyQ+SSJURzYiIiM5ISIi QyQ+SSVydW5zRzYiNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSlleHBvcmFuZEdGJTYmJCIiJSEiIiIiIkkiVEdGJS1JJ05vcm1hbEc2JEYpSShfc3lzbGliR0YlNiRGMSQiIiZGMC9JImpHRiU7RjEiJStdRjA= QyQ+SSV0b3BzRzYiNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSVjZWlsRzYkRilJKF9zeXNsaWJHRiU2IyYmSSVydW5zR0YlNiNJImpHRiU2I0kiVEdGJS9GNDsiIiItSSVub3BzR0YpNiNGMiEiIg== QyQtSSxEYXRhU3VtbWFyeUc2IjYjSSV0b3BzR0YlIiIi Guess: The final population sizes follow a log normal distribution (could also be Gamma?) QyQ+SSNtdUc2Ii1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSSdNZWRpYW5HRiU2I0kldG9wc0dGJSIiIg== QyQ+SSZzaWdtYUc2Ii1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmSSNtdUdGJSIiIi1JI2xuR0YoNiMtSSVNb2RlR0YlNiNJJXRvcHNHRiUhIiJGLg== QyQ+SSRwbDFHNiItSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiYtSSRQREZHRiU2JC1JKkxvZ05vcm1hbEdGJTYkSSNtdUdGJUkmc2lnbWFHRiVJInRHRiUvRjQ7LUkkbWluR0YpNiNJJXRvcHNHRiUtSSRtYXhHRilGOS9JKnRoaWNrbmVzc0dGJSIiIy9JJmNvbG9yR0YlSSRyZWRHRiUhIiI= QyQ+SSJIRzYiLUkqSGlzdG9ncmFtR0YlNiVJJXRvcHNHRiUvSS9mcmVxdWVuY3lzY2FsZUdGJUkpcmVsYXRpdmVHRiUvSS9hdmVyYWdlc2hpZnRlZEdGJSIiJSEiIg== QyQtSShkaXNwbGF5RzYiNiM3JEkiSEdGJUkkcGwxR0YlIiIi QyQ+SSJTRzYiNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQ3Iy1GKDYkJiZJJXJ1bnNHRiU2I0kia0dGJTYjSSJ0R0YlL0YyOyIiIiwkLUklbm9wc0dGKTYjRjAjRjciI10vRjQ7RjdJIlRHRiUhIiI= QyQtSSl0ZW1wcGxvdEc2IjYjSSJTR0YlIiIi Logistic growth QyQ+SSJURzYiIiNJISIi QyQ+SSVydW5zRzYiNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSlsb2dpcmFuZEdGJTYnJCIiJSEiIiIlKzciIiJJIlRHRiUtSSdOb3JtYWxHNiRGKUkoX3N5c2xpYkdGJTYkRjIkIiNEISIjL0kiakdGJTtGMiIlK11GMA== QyQ+SSV0b3BzRzYiNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSVjZWlsRzYkRilJKF9zeXNsaWJHRiU2IyYmSSVydW5zR0YlNiNJImpHRiU2I0kiVEdGJS9GNDsiIiItSSVub3BzR0YpNiNGMiEiIg== QyQtSSxEYXRhU3VtbWFyeUc2IjYjSSV0b3BzR0YlIiIi Guess: The final population sizes follow a normal distribution QyQ+SSRwbDFHNiItSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiYtSSRQREZHRiU2JC1JJ05vcm1hbEdGKDYkLUklTWVhbkdGJTYjSSV0b3BzR0YlLUkyU3RhbmRhcmREZXZpYXRpb25HRiVGNEkidEdGJS9GODstSSRtaW5HRilGNC1JJG1heEdGKUY0L0kmY29sb3JHRiVJJHJlZEdGJS9JKnRoaWNrbmVzc0dGJSIiIyEiIg== QyQ+SSJIRzYiLUkqSGlzdG9ncmFtR0YlNiVJJXRvcHNHRiUvSS9mcmVxdWVuY3lzY2FsZUdGJUkpcmVsYXRpdmVHRiUvSS9hdmVyYWdlc2hpZnRlZEdGJSIiJSEiIg== QyQtSShkaXNwbGF5RzYiNiM3JEkiSEdGJUkkcGwxR0YlIiIi QyQ+SSJTRzYiNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQ3Iy1GKDYkJiZJJXJ1bnNHRiU2I0kia0dGJTYjSSJ0R0YlL0YyOyIiIiwkLUklbm9wc0dGKTYjRjAjRjciI10vRjQ7RjdJIlRHRiUhIiI= QyQtSSl0ZW1wcGxvdEc2IjYjSSJTR0YlIiIi JSFH