Bycatch of dolphins data, zero inflated and under-dispersed - what next?

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.


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You had me at dolphin. What is a bycatch? Yes, I have also heard those alternative are better suited for overdispersion.

I personally don't use these model much and I am curious how you are determining under-dispersion? I typically thought about over dispersion as variance greater than distribution mean. You mention many zeros, which intuitively may signal underdispersion?
Sorry, I should have been more clear - where I refer to bycatch I am talking about the unintentional capture of dolphins in fishing nets. I read that dispersion can be assessed using the ratio of residual deviance to residual d.f. If it is less than one it indicates underdispersion and if greater than one it indicates overdispersion.

Many thanks