How do I handle a normal distribution with peaks at each end? This question may appear to be quite odd, but the following explanation should make it a little more comprehensible. 
I work on the analysis of population trends of birds. In our team, we work with a simple trend index, that is calculated as 
$$\frac{\rho_t - \rho_{t+1}}{\rho_t + \rho_{t+1}} = \frac{(\mbox{density at time }t) - (\mbox{density at time }t+1)}{(\mbox{density at time t })+ (\mbox{density at time }t+1)}$$ 
When the population size does not change, the index value is $0$. When the population decreases, the index is negative; when the species goes extinct, the index value is $-1$. An increasing population yields positive index values, with the extreme of $1$ when the species colonises a formerly uninhabited area. We calculate this index for a large number of monitoring plots. For abundant species the index values of all plots together yield a nice, more or less bell-shaped curve that is convenient for further analysis. For less abundant species, however, the number of plots where it disappeared, and the number of newly colonised plots can be large leading to a three-modal curve with modes at $-1$, $0$ and $1$. The rarer the species, the higher get the peaks at $-1$ (extinction) and $+1$ (colonisation) and the flatter gets the bump in the middle. Analyses of such data (e.g. in various types of regression analysis) are difficult since I am not aware of a statistical distribution that can describe them.  
I know that this is not a very precise question that can easily be answered, but I'd be thankful for any advice how to treat data like this or how to calculate a more "user-friendly" trend index for two points in time.
 A: I suspect that the problem here pivots on the feeling that the distribution should be normal (Gaussian), or more nearly so. But why precisely? It's nowhere an assumption of regression (for example) that marginal distributions, the distributions of any of the variables input, be normal. 
The index is what it is: it sounds well defined and natural and easy for people in your field to think about and those are real positives. The emphasis should be on adapting any analysis method to respect the way it behaves, not on changing the data in anticipation that they are in the wrong form or have the wrong distribution. 
For example, let me guess that you might to try modelling this index as a function of other variables. Then it's important that any model not predict impossible values, outside $[-1,1]$. For that (index + 1)/2 recasts the range to $[0,1]$ and then you can apply logit models of an appropriate kind. (In practice, that means a special kind of generalised linear model that supports logit link, binomial family [sic] and robust (Huber-White) standard errors.) 
A: If you only need a distribution to describe your data, you might be able to use a mixture of two deltas and one normal distribution, or possibly a truncated normal or circular normal distribution.
$$\pi_{-1} \delta(x + 1) + \pi_0 \mathcal{N}(x; 0, \sigma^2) + \pi_1 \delta(x - 1)$$
