GLM for a time VS proportion relationship

Hi everyone,

I would like to verify a relationship between two variables : "mean time spent foraging" and "proportion of grassland" in the individual home ranges for different individuals (id).

I have tried linear mixed models with log(y) and id as a random effect but my residuals do not follow a normal law. A smoothing with a Generalised Additive Model always give me a linear relationship so I am probably missing something. Since my dependent variable do not include integers, I can't do a GLM with a poisson distribution.
Do you think there is another distribution I can try with this data (time x proportion) ?

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