For a mixed model, does the temporal correlation should be checked for each site (random effect)? When a model fitted in a dataset with multiple time series from different sites, does the temporal correlation should check for each site? And if some sites show auto-correlation but some don't, what's that mean and how to deal with? Thanks.  
Here is details. I fitted a model like:
m1 <- gam(Abundance ~ s(Temperature) + s(Temperature, by=Site, m=1) + 
            s(Site, k=11, bs="re") + s(Year),
          method="REML", family=binomial(link="logit"),
          weights=Trap, data=c2)

I checked the auto-correlation plot for residuals for each site. Most of sites don't show residual problems, but some sites do. I added a residual auto-correlation structure, for example:
m2 <- gamm(Abundance ~ s(Temperature) + s(Temperature, by=Site, m=1) + 
             s(Site, k=11, bs="re") + s(Year),
           correlation=corAR1(form = ~ Year|Site),
           method="REML", family = binomial(link="logit"),
           weights = Trap, data=c2)

or
m3 <- gamm(Abundance ~ s(Temperature) + s(Temperature, by=Site, m=1) + 
             s(Site, k=11, bs="re"),
           correlation = corAR1(form = ~ Year|Site),
           method="REML", family = binomial(link="logit"),
           weights = Trap, data=c2)

Always have a message = iteration limit reached without convergence (10) 
Any advices would be appreciate.
 A: If there is some structure in the model residuals, even if it is at just one or two of the Sites, will lead to the test statistics being anti-conservative (p-values will be somewhat smaller than they should be, confidence intervals somewhat narrower) because they assume the observations are conditionally independent.
It doesn't matter that some Sites show no residual autocorrelation but you're estimating an AR(1) parameter for them as the parameters for those sites will most likely be very small. The only price you pay is in estimating these potentially redundant parameters but if we're being honest, the autocorrelation in those Sites is probably not exactly zero so the model is a reasonable reflection of what the data is telling you.
You might look to see if the basis size for s(Year) is sufficiently large; increase k for this smooth and refit and see if that can model the autocorrelation so you don't need the AR(1). You might also change s(Year) to s(Year, by = Site) to see if you have different trends per Site which might mean you aren't modelling all the trend in some Sites which may be showing up as residual autocorrelation.
A: I tried both increase k for s(Year)(the k value is enough) and s(Year,by=Site),there still have residual autocorrelation:
E1=resid(m1)

acf(E1)


I delete the term s(Temperature, by=Site, m=1)and add AR1 as:
m2 <- gamm(Abundance ~ s(Temperature) + s(Site, k=11, bs="re") + s(Year), correlation=corAR1(form = ~ Year|Site), method="REML", family = binomial(link="logit"),weights = Trap, data=c2)

It seems works well. But given the AIC, allowing a random effect of Temperature by site fits better than only a global function. However, there always have convergence problem when fitted with a correlation structure. Then I tried a GLMM:
m4<-glmmPQL(Abundance~Tem_avg+Tem_dev+Year,random = ~1 + Tem_dev | Site,correlation = corAR1(form = ~ Year|Site),family = binomial(link="logit"),weights = Trap, data=c2)

It seems works well. I also plot the residual against all continuous covariates in the model:


It seems the resulting panels do not indicate obvious non-linear patterns (maybe a little non-linear pattern for Year,I'm not sure whether I could neglect it or not).
