Do non-human animals suffer from insomnia?
Insomnia most certainly occurs in other animals. One interesting example is the case of insomnia in Drosophila melanogaster. In this study 3 day old male and female flies that demonstrated reduced sleep time were crossed together over 60 generations to create flies insomnia-like (ins-l) that sleep less than 60 minutes a day compared to 800 min a day in their control flies, which are generation 0 (Fig 1B). These (ins-l) flies exhibited difficulty initiating sleep, maintaining sleep and showed cognitive impairment. They also had difficulty maintaining balance with elevated levels of triglycerides, cholesterol and fatty acids and although their core molecular clock remained intact, ins-l flies lost their ability to sleep in constant dark. Using whole-genome transcriptome profiling using Affymetrix high density nucleotide microarrays, these flies are compared against the control flies and many genes were over-represented (changed in expression compared to control flies) which are involved in metabolism, neuronal activity, behaviour and sensory perception with changes in lipid metabolism and synaptic transmission being consistent with increased adiposity and learning defects in flies. In addition, these ins-l flies were hyperactive and hyper responsive to environmental simulations and the over-representation of genes associated with sensory perception suggests that hyperarousal in insomnia may be attributed to increased activity in the sensory system and that these processes are under genetic control. Two genes in particular Filamin-A (Cheerio) and Malic enzyme (Men), which were found to be differentially regulated (changed in expression) in files are also modulated (changed) by sleep loss in humans, showing that the same processes are involved/get effected in humans and flies during insomnia.

The research above clearly demonstrated the genetic regulation and effects of insomnia in other animals and shows that its not just a human dysfunction.