Logistic Growth models for Count Data I have a dataset of monthly ridership figures by transit route from 2007 to 2015.  I am analyzing this data in R.  When I go to predict on a new dataset with step increases in trips (ie 1,2,3,etc.) using any PLM, GLM, NLMER or GEE/GENLIM analysis my ridership predictions all increase linearly or exponentially.  
I understand why, but am looking to find an alternate method where the ridership increases on a logistic scale.  Because I am dealing with count data, I could transform the data to make all ridership values between 0 and 1.  However any predicted cases where there is a large increase in this trips variable past the largest value in my dataset would see an extremely small change in ridership.
Is there a logistic approach out there that can be applied to count data or is there a way to allow that the maximum could be higher (ie delaying the decreasing rate of ridership given higher values of trips (for example))? 
 A: What you seem to be describing is a dynamic growth model. Not sure regression is the appropriate method. Most regression is correlational--it susses out the relationships between different variables at a single point in time. You are trying to predict the future value of a dynamic process, which typically requires a simulation model. If you are determined to use regression, then you need to figure out the causal factors driving your increase in ridership, something like:
OLS1=lm(Ridership~Pop+Employment,data=yourdata), and then you can include your predictions of future pop and employment to estimate future ridership. 
If you are just looking for an equation that will permit you to make your growth rate inversely proportional to your ridership, like so:

Then something like: 1/1+exp(Ridership) ought to do the job.
Full procedure for fitting such a sigmoid curve to your data can be found here:
http://www.stat.wisc.edu/~ifischer/Intro_Stat/Lecture_Notes/APPENDIX/A4._Regression_Models/A4.3_-_Logistic_Growth.pdf
