A temporal model of territorial defence with antagonistic interactions

Territorial behaviour is an important part of the lives of many animals. Once a territory has
been acquired, an animal may spend its entire life on it, and may have to repeatedly defend it from
conspecifics. Some species make great investments in the defence of a territory, and this defence
can be costly, in terms of time, energy and risk of injury. Time costs in particular have rarely
been explicitly factored into such models. In this paper we consider a model of territorial defence
which includes both population dynamic and time delay elements, building upon recent advances
in time constraint models. Populations may divide into two distinct types, where one type makes
no effort to control territories. We shall call this type nomads, and the other type territorials.
Here the territory owners must divide their time between patrolling and foraging, and this
balance is their only strategic decision. We show how to find the evolutionarily stable patrolling
strategy and the population composition of territorials and nomads, and consider some examples
demonstrating key situations. We see that both time constraints and population density pressure
are crucial to influencing behaviour. In particular we find cases with both territorial individuals
and nomads where a mixed, either pure or both pure patrolling strategies are evolutionarily
stable. In different conditions either nomads or territorials can be absent, and indeed for a
significant range of parameter combinations the population can exhibit tristability, with three
distinct ecologically stable population compositions: with both nomads and territorials, only
nomads or only territorials.