What is tearing up/eating lawn roots?

I live in Cuenca, and have a large green lawn. Every morning, I find patches in the lawn that have been torn up, and discarded grass/roots lay on the lawn in clumps. Something is pulling out the grass and eating the roots at night.

I was told by the locals the problem was rats, so I put out rat traps, but those were ignored and the problem has continued. What is the culprit, and how do I put an end to it?