A Free Boundary Problem Arising from a Bistable Reaction-Diffusion Equation.
Abstract
The mathematical equation studied here has been considered as a model for population genetics, combustion, and nerve conduction. A common feature to all of these phenomena is the existence of traveling wave solutions. These may correspond, for example, to the spread of an advantageous gene through a population or the propagation of electrical impulses in a nerve axon. Another common feature is the existence of a threshold phenomenon. In the nerve, for example, a small initial stimulus will not trigger an impulse. If the initial stimulus is greater than some threshold amount, however, a signal will propagate down the axon. In this case, the signal quickly assumes a fixed shape and travels with constant velocity. Physiologically, it has been demonstrated that this shape and velocity is independent of the initial stimulus, as long as the stimulus is above threshold. In this report, we demonstrate that the mathematical model under consideration does indeed exhibit a threshold phenomenon. We also study how initial stimuli evolve into traveling wave solutions.
Document Details
- Document Type
- Technical Report
- Publication Date
- May 01, 1981
- Accession Number
- ADA100566
Entities
People
- David Terman
Organizations
- University of Wisconsin–Madison