Wavetrain selection following predator invasions in oscillatory reaction-diffusion systems.
by
Sandra Merchant
University of British Columbia
Coauthors: Wayne Nagata (University of British Columbia)

Periodic travelling waves, also known as wavetrains, are known to evolve behind invasion fronts in oscillatory reaction-diffusion models for predator-prey systems. Mathematical theory predicts that for a given set of parameter values there is in fact a family of possible wavetrain solutions and in a particular predator invasion a single member of this family is somehow selected. Sherratt (1998) has studied this selection mechanism, using the Normal Form approximation that is valid for such models near the Hopf bifurcation in the local system. However, away from this Hopf bifurcation the predictions from the Normal Form lose accuracy. We conjecture a more general selection criterion that retrieves the prediction from the Normal Form system, but that applies to the full (non-reduced) predator-prey system and that depends on the properties of the wavetrains for the full system and hence retains accuracy away from the Hopf bifurcation. We illustrate how to apply this selection criterion using three sample oscillatory reaction-diffusion models from the literature on predator invasions. The selection criterion does indeed provide more accurate predictions for these models than the criterion based on the Normal Form, but does eventually lose accuracy as well. We therefore conclude with future directions for work on this problem.

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Date received: April 23, 2008