lmer, random effect with four levels

Dear Amir,
the "place" random effect in your dataset has only four levels, and they are nested within plants. The error which is returned states that the variance covariance matrix is singular, meaning that variation in intercept, slope and their covariance cannot be fitted. If you fit the random effect as (1|place), you will read that the random effect variance is effectively zero.
It does not make sense to include place both as fixed and as random, as in you second to fourth models. Since the number of levels of the random effects is very low, you could even just model them as fixed, using glm. The idea of random effects is that they are a representative sample from a large population, which is hard to defend with just two levels sampled per group!
More importantly, the distribution of your data does not look poisson at al. There are many zero observations. I would convert the counts to presence/absence data and analyse those as bernoulli probabilities. That works to some extent:

b<-ifelse(Predators.Allspecies.>0,1,0)
model<-lmer(b~julian.date+offset(log.time)+(1|place),family=binomial)

I've added you models to your post, so other visitors to this blog can also read and respond to your question,
all the best, Tom