Eww...I love some proverbs
"It is better to feed one cat than many mice."
"When the cat's away the mice will play"
"A Cat is a Lion to a mouse."
"Mice won't dance in a cat's doorway."
We too live with a pasture and we do not have mice, neighbors on either side, no cats own these neighbors and they have mouse problems. Our two indoor cats patrol house and garage where bird seed, is stored. Kip-purr and Girlie keep our home rodent free. Our Miniature Wire haired Dachshund catches mice and voles (eww inserted here) in the garden area...he's a terrific outside mouser. It is said that if mice enter into an area and sense that cat's are there they move on...in our experience that certainly seems to be the case.
There is one thing that you can be certain of, if you see one mouse and or droppings there are many mice in the area or will be within 3 weeks. One mouse gestates in 17-21 days giving birth from 3-14 per litter. Females can breed @ 5 weeks, males the same. Having 5-10 litters a year....
Isn't it interesting, at least to me....that there is now gray area when it comes to cats. It seems either people like/love cats or they hate them. I understand allergies, I also understand generalities but that has been my observation.
Mice dart across open floors...their highway's are along baseboards....place some traps along their highway's and where there is evidence of where they party. I would suspect that you already have a population considering the damage done in such a short time.
Like my Mom used to say it's no shame to have a mouse, it's a tragedy to provide a home for them. So your doing good girlfriend in getting busy to carry those buggers out in body bags. However since you've previously had a cat...it would be wonderful to provide another kitty a loving home and it will repay you with keeping your home free of these filthy rodents.