GAM: Find a good distribution for the monthly data sums? I am new in the GAM modelling. I would like to find a family, that will fit my response variables. I am using the sums of monthly counts of beetles, collected from the beetle traps in ~ two weeks interval (this can vary between traps and years) and from diverse locations across Germany. My datasets contains some zeros, but not too many. Also, as I have sum of counts, my data are always positives and integer. As I am moving from counts (discrete), are now my sums continuous ones?
Shortly, my dataset contains some zeros (~10% of data), and also extreme values, where many traps have very low counts. Here is how my monthly sums looks like:

I have found that Tweedie distribution can account for existing zeros and is often used in GAM models in similar studies. But when applied in my data, I almost find a perfect fit, but weird pattern exactly at low/zero values:

I am not sure, how can I account for this data chunk in my family? I have tried different combination of a, b, and theta parameters, and tested several families in mgcv package (as I will further use bam). nb family has a good fit, but again have a weird pattern at low values. Maybe you have some suggestions how can I fit my data better? Thank you!
Here is my code, using gam(y~1) to only fit the y distribution, without any predictors:
m <- gam(count_sum ~ 1,dat, family = tw(link = 'log', theta = 1.85, a=1.82,b=1.99)) 
gratia::appraise(m)

I wonder, if this can be done only by adjusting the parameters within family, or should I move to completely different family? Thank you for your thoughts.

My study design is very similar to Irregular time series data  including long-term trends, and spatially varying (e.g. share of the forests in each trap surrounding). Following @Gavin Simpson comment, I expect that the trap counts depends on location (XY) and time (variation betweeen months, between years). As suggested by @Gavin Simpson, I should move from using a single distribution to use different distribution for each trap? How can that be implemented?

While Stephen's answer explains and discusses the potential zero-inflation in your data, there are several other considerations including your use of the tw() family and the underlying assumptions of the model you fitted.
Firstly, the tw() family allows you to not specify any of the parameters and have the underlying algorithm find the optimal power parameter for the Tweedie distribution fitted.
The other arguments a and b are only used if you want to limited the search interval over which values of the power parameter are considered. It doesn't make much sense to limit the searched interval too much like you are doing without good justification.
I think it is highly unlikely that all your sampling locations are so homogeneous in space and time that you can model them with a single distribution, Tweedie or otherwise.
You mention monthly counts (I presume this is where the sums come from - you are summing the two-weekly samples into monthly counts? - some additional detail on your setup, how you are summing etc would be useful) so perhaps the summed counts vary over time reflecting some phenological process? Are your traps all in the same location such that it is reasonable to consider them all to have the same expected count? If the environments around your traps differ such that we might expect more or fewer beetles in general, or of certain species, then using a single distribution to model all the summed counts would be too restrictive.
If it really is reasonable to model the observations with a single Tweedie (and summed counts seems to be a reasonable justification for the Tweedie, assuming that there are many such counts in each sum?) then you will likely be better off fitting the Tweedie using the Tweedie package as it considers a wider family of Tweedie distributions where the tw() family is limited to powers in the range 1–2, but zero-inflated or hurdle models would also be appropriate as Stephen suggested and consider zero-inflated or hurdle versions of the negative binomial as well as the Poisson.
