Why do animals have territories?

In many animal species some individuals defend territories and the food or mates that they contain against others. Defending an entire area is complicated, potentially dangerous and requires a lot of time and effort. Why is it still advantageous for animals to do it and how could this behaviour have evolved in the first place?
The costs and benefits of defence and thus the scenarios in which it pays to defend will vary greatly depending on whether owners prevent other animals from taking over their territory, from stealing resources or from changing the territory borders. A failure to make these distinctions explicit has probably contributed to the slow progress in the field.
Using simulation models my co-authors and I find that individuals can indeed evolve to defend the resources in their vicinity. Whether they do so, however, very much depends on how exactly interactions between them work. If they do defend and fighting is costly enough we find in that in the next step distinct contiguous territories can evolve, however, again dependent on very specific details in the assumptions of the model.
We further find that under naive assumptions it should nearly always be good for territory owners to try to steal from their neighbours which again can make defence too expensive to be worthwhile. However if potential thieves can change their behaviour depending on how likely it is that they will be attacked they will evolve to become very cautious and defence can be maintained.