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# Long Term Dynamics for Two Three-Species Food Webs

## Abstract

In this paper, we analyze two possible scenarios for food webs with two prey and one predator (a food web is similar to a food chain except that in a web we have more than one species at some levels). In neither scenario do the prey compete, rather the scenarios differ in the selection method used by the predator. We determine how the dynamics depend on various parameter values. For some parameter values, one or more species dies out. For other parameter values, all species co-exist at equilibrium. For still other parameter values, the populations behave cyclically. We have even discovered parameter values for which the system exhibits chaos and has a positive Lyapunov exponent. Our analysis relies on common techniques such as nullcline analysis, equilibrium analysis and singular perturbation analysis.

## Author Bio

The research for this paper was completed at an REU program at the University of Nebraska-Lincoln in the summer of 2002. Currently in the fall of 2003, I am a senior at the University of Nebraska-Lincoln where I am double majoring in mathematics and meteorology/climatology. I will graduate in May of 2004. I plan to attend graduate school in applied mathematics.

This research was done during the Summer 2002 REU at Central Michigan University under the direction of Dr. Ken W. Smith. This was after my sophomore year at the University of Missouri - Rolla. My mathematical interests are group, graph, and number theories. Outside of academia, I love to jitterbug and lindy hop and am an active member (currently president) of the Ballroom Dancing Club at my university. In addition, I am slowly but surely learning the art of Tai Chi Chu'an. In what spare time I have left, I am also working on my first reading of the Robert Jordan Wheel of Time series.

This work was completed at a summer REU program at the University of Nebraska-Lincoln. At the time, I was a senior Applied Math major at Kent State University. This summer REU was such a positive experience, it inspired me to pursue a higher degree in mathematics. I am currently working on my Master's Thesis on the bifurcation of periodic structures in liquid crystal films at Kent State University.

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