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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

An Integrodifferential Equation Modeling 1-D Swarming Behavior

Leverentz, Andrew 01 May 2008 (has links)
We explore the behavior of an integrodifferential equation used to model one-dimensional biological swarms. In this model, we assume the motion of the swarm is determined by pairwise interactions, which in a continuous setting corresponds to a convolution of the swarm density with a pairwise interaction kernel. For a large class of interaction kernels, we derive conditions that lead to solutions which spread, blow up, or reach a steady state. For a smaller class of interaction kernels, we are able to make more quantitative predictions. In the spreading case, we predict the approximate shape and scaling of a similarity profile, as well as the approximate behavior at the endpoints of the swarm (via solutions to a traveling wave problem). In the blow up case, we derive an upper bound for the time to blow up. In the steady state case, we use previous results to predict the equilibrium swarm density. We support our predictions with numerical simulations. We also consider an extension of the original model which incorporates external forces. By analyzing and simulating particular cases, we determine that the addition of an external force can qualitatively change the behavior of the system.

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