Want to regress (statistically), but can it be done?

I have population size over time with many gaps (the unfortunate reality of many biological data sets!): there are only 12 points over 37 years.
The goal of the analysis is try to figure out what has influenced the population trajectory (climate, human use, predator numbers). My plan was to identify a few candidate models a priori (under competing hypotheses), do several multiple linear regressions, and use AICc to select a model that has the most support. But, I’m leery about using regression with so many gaps in the data, and I’m not sure what is the best response variable under this scenario anyway.
2 questions:
Does regression seem appropriate with the many gaps in the data (most of the independent variables, (i.e. weather, human use) are available for every year!)?
For the DV, I’d like to use a growth rate, to get away from actual “counts” (which would probably require a GLM count-data model), but is it OK to use growth rate if I have up to 15 year gaps in the data?
Any thoughts on this would be appreciated!
Thanks for your thoughts!

Yes, you are correct, there are 12 points, showing a declining population over 37 years.

I have certainly seen published regression analysis with <20 data points in the dependent variable (Green 1997, Christensen et al. 1996). A general rule of thumb I've learned is to keep the number of variables at least 5 times less than the number of data points in your DV (i.e. with 10 points, your model should have only 2 variables).

If regression isn't appropriate, do you know any alternative ways to analyse this question? I could lump data into 5-yr averages and compare groups in an ANOVA?