I am investigating the factors that influence the bycatch of dolphins using a poisson GLM. My response variable is bycatch rate per unit effort and varies from 0 to 0.125. However, my response is very zero-inflated (2304 of 2436 observations are zero). I have three response variables (two categorical (5 categories each) and one continuous). My model results show significant under-dispersion (21.156 residual deviance on 2434 degrees of freedom). I have tried every possible transformation on my response variable and this has done nothing. I considered using a negative binomial or zero-inflated model but have read that these are better for over-dispersed data. I would greatly appreciate advice on anything else I could try.