Time Budgets and Territory Size: Some Simultaneous Optimization Models for Energy Maximizers

Abstract

A set of optimization models in two variables of choice, territory size and time spent patrolling for intruders, is presented for energy maximizers. Models vary in the curvilinearity of the relationship between territory circumference and both intrusion rate and cost of expelling a single intruder. Models are analyzed both with and without constraints; constraints are on processing rate and on the time spent patrolling, feeding and actively defending. The models all include the concept of “intruder equilibrium,” an equilibrial density of intruders in a territory resulting from a balance between intrusion rate and expulsion by the defender. This equilibrial density can be considered a measure of territorial exclusiveness. The two-variable models predict effects on territory size and patrol time of variation in food density, intrusion rate, costs of expelling a single intruder in energy and time, food-consumption rate of an intruder, area of detection while patrolling, total time available for territorial and feeding activities, time to eat a unit of food energy, energy cost of patrol per time, and processing-rate capacity. With increasing intruder rate, optimal territory size usually decreases, whereas optimal patrol time behaves much more irregularly. With increasing food density, optimal patrol time usually decreases, whereas optimal territory size behaves irregularly. In particular, when intrusion rate and expulsion costs accelerate sufficiently with increasing territory size and no constraints exist, the higher the food density the smaller the optimal territory size. When food density is large enough for a constraint to be effective, the opposite relation can hold and will always hold for a processing constraint. When a particular parameter changes, optimal territory size and optimal patrol time may covary or one may increase while the other decreases, depending on the parameter and model. A new set of one-variable models is suggested by the two-variable models; models optimizing patrol time while holding territory size constant could correspond to a tightly packed system of territories initially determined by settlement patterns. A unified onevariable analysis suggests that how food density affects territory size when patrol time is constant depends upon whether a constraint is operating: Provided that invasion rate does not vary with density of intruders on the territory, time minimizers and constrained energy maximizers decrease territory size with increasing food density; unconstrained energy maximizers do the opposite. The addition of a second optimization variable to a one-variable model can change qualitative predictions about variation in particular parameters ( e.g. , food density) and can increase the number of parameters predicted to affect optimal territory size and patrol time.