Interval Analysis and its Applications to Optimization in Behavioural Ecology

A comparison of the interval and fixed-point results will be done as well as analysis of parameter intervals and their constraints, root approximations, and applications to the model.

Key Words: single variable optimization, interval analysis, behavioural ecology, foraging

Foraging models in general study two basic problems of a forager: which food/prey items to consume and when to leave an area containing food (a resource patch). This paper will concentrate on the latter as an optimization problem. The paper discusses the framework of foraging models and also explains foraging and optimality models through the three main components of decision, currency, and constraint assumptions. Given the structure of a foraging model, it is easy to frame a model examining the foraging of a single animal over a collection of distinct resource patches. I have taken the model along with its decision, currency, and constraint assumptions from Wilson (2000) (in references) so the construction of the model's equations, its origins, and an analysis and extensions of the model in C can be found in his book. The rules of the forager in the model are that the animal stays for a fixed time before moving to a new resource patch, time is discrete, and patch resource values (biomass, energy, etc) grow logistically. The decision assumption lies in the determination of the fixed time value. Assuming behavioural and evolutionary mechanisms drive foragers to optimize the time spent on each patch. This assumption implies that they will stay long enough to optimize the rate of resource consumption.

The mathematical conclusion, explained in the paper using the model's structure is that the optimal time to leave a patch is when the expected rate of resource return decreases to the average rate of return of a new patch (a statement of the marginal value theorem for foraging). This result then allows an optimization problem to be constructed.

IA's applications to the foraging model include adding more realism to the model through deterministic variability as well as allowing the ecologist to select which parameters are to be uncertain, their range, or different beginning time intervals. The variability is important due to the stochastic nature of ecological process and IA offers an alternative to average state approximations.