It's all about lighting, age, and breed genetics. A drake will not increase production to year round nor should they be fed "layer" feed.
Cat food is formulated for cats.. very high in protein.. and not the right mix of nutrients and minerals.. maybe good for emergency purposes and if already on hand.. but not what I would call a wise choice long term.. if you wanna up protein.. use a gamebird or turkey starter.. but flock raiser is a good choice (with oyster shell on the side).
All birds will quit laying while they molt and and use that energy to replace feathers.. usually age and season related. These breaks in laying are required for decent health.
With chickens.. the way I keep production up year round is by adding a couple young layers each year that won't be molting come that fall/winter when the older ones are.
Also, you "might" be able to preserve duck eggs from higher production times by "water glassing" (hydrated lime soak).. simple clean unwashed refrigeration gives at least equal result to what I've seen reported (and even safe for 4-6 months, though won't quite whip up) but duck eggs are usually messier so might not last as long under refrigeration.
Here is a link about egg production from a fairly valuable resource of information (including breed comparisons) if you look under some of the drop down tabs (even though it's a hatchery)..
https://www.metzerfarms.com/maximizing-egg-production.html
Hope this helps some.