**Researcher:** Justin Tung (jt96@cornell.edu)**Instructor:** David Schwartz (dis@cs.cornell.edu)

## Contents

- Abstract
- Interval Analysis
- Behavioural Ecology and the Foraging Model
- Methodology
- Summary Results
- Paper and Software
- References

# Overview

Interval Analysis is a means of representing uncertainty by replacing single (fixed-point) values with intervals. In this project, ** interval analysis is applied to a foraging model in behavioural ecology**. The model describes an individual foraging in a collection of continuously renewing resource patches. This model is used to determine the optimal residence time of the forager in a resource patch assuming the forager wants to maximize its rate of resource intake. Before applying interval analysis, fixed-point (non-interval) optimization will be done to serve as a basis. Certain parameters in the mode will then be replaced with intervals and interval-based optimization conducted.

# Abstract

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

# Interval Analysis

The paper covers only a brief outline of **Interval Analysis**** (IA) ** and the math associated with IA that is needed to understand the topics covered and assumes some prior knowledge of IA and Matlab though it is possible to understand the applications to the foraging model without that knowledge. For a formal mathematical introduction and in depth coverage of concepts see Schwartz (1999) or Moore (1966) listed in the references. Interval analysis is a deterministic way of *representing uncertainty in values by replacing a number with a range of values* (Schwartz 17). Fixed-point analysis is simply analysis using non-interval values where there is no uncertainty in the values. As a result, IA uncertainty concepts can be used to model varying biological parameters in the ecological model and also to frame fixed-point results. One website providing an excellent gateway to interval analysis resources and information is at http://www.cs.utep.edu/interval-comp/main.html

# Behavioural Ecology and the Foraging Model

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.

# Methodology

The optimization problem was approached in two ways. One way was a fixed point analysis of the problem using the *bisection method* after some graphical analysis of the functions involved. The second way was an IA approach using the **interval Newton’s method**. The results and details of these two methods and their comparisons can be found in the paper. A **stability analysis** was also conducted on the model, which was testing the limits of the uncertainties of the variables before the model “broke down”.

# Summary Results

The values obtained from both optimization methods corresponded well since error bounds derived from the error between the fixed-point roots and the endpoints of the optimal residence time intervals shows relatively small differences from each other. The use of IA to simulate a range of parameter states in the system was successful since the percent uncertainties yielded interval optimal residence times. As a result, IA allows the ecologist to vary parameters of the model within certain limitations established by stability analysis and obtain a range of optimal residence times a forager might choose. The percent uncertainty limitations on parameter variation obtained from the stability analysis are high, which is good since it allows a large amount of model variability.

*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.

# Paper

Interval Analysis and its Applications to Optimization in Behavioural Ecology, January 2002 HTML or download PDF report

## Software

Matlab Source files are available on GitHub along with the paper. See the paper regarding descriptions and functions of the file. Files are also available in a zip format. List of files:

Text User Interface | Support Functions | Data Analysis | Numerical Algorithms |

main.m dispara.m disparafp.m disparaia.m fpOptimizeMain.m intOptimizeMain.m selectP.m selectPIA.m | gain.m gainPrime.m gawMax.m newInterval r.m rootFunc.m rootPrime.m | sa.m ga.m fpOptimize.m intOptimize.m | bisection.m intNewton.m UnitedNewtonStep.m |

**II. Project Files**

The language used in this project was Matlab version 5.3 with an add-on toolbox called INTLAB programmed by Siegfried Rump (2001). Refer to http://www.ti3.tu-harburg.de/~rump/intlab/index.html for additional information on the INTLAB toolbox. For this paper, the INTLAB toolbox was used to provide interval data structures, implementation of interval arithmetic and interval-valued functions, as well as basic functions for radius, midpoint, and intersection interval functions in Matlab.

# References

Hart, John C. “Ray Tracing Implicit Surfaces.” Notes for Advanced Surface Modeling. http://graphics.cs.uiuc.edu/~jch/cs497jch/rtis.pdf (8 Nov. 2001).

Hansen, E. R. “Global Optimization Using Interval Analysis: The One-Dimensiona Case” Journal of Optimization Theory and Applications 29 (1979): 331-344.

Krebs, John R. and N. B. Davies. An Introduction to Behavioural Ecology. 2nd ed. Sunderland: Sinauer Associates, Inc., 1987.

Kulisch, U., R. Lohner, and A. Facius., eds. Perspectives on Enclosure Methods. New York: Springer-Verlag, 2001.

Moore, Ramon E. Interval Analysis. Englewood Cliffs: Prentice-Hall, Inc. 1966.

Neumaier, Arnold. Interval Methods for Systems of Equations. New York: Cambridge University Press, 1990.

Rice, John R. Numerical Methods, Software, and Analysis. 2nd ed. New York: Academic Press, Inc., 1993. Rump, Siegfried M. “INTerval LABoratory Version 3.” INTLAB http://www.ti3.tu-harburg.de/~rump/intlab/index.html (14 Dec. 2001).

Schwartz, David I. Deterministic Interval Uncertainty Methods for Structural Analysis. Diss. State University of New York at Buffalo, 1999.

Stephens, David W. and John R, Krebs. Foraging Theory. Princeton: Princeton University Press, 1986).

Van Loan, Charles. Introduction to Scientific Computing. 2nd ed. Upper Saddle River: Prentice Hall 2000.

Wilson, Will. Simulating Ecological and Evolutionary Systems in C. Cambridge: Cambridge University Press, 2000.