This article is concerned with pointwise growth and spreading speeds in systems of parabolic partial differential equations. Several criteria exist for quantifying pointwise growth rates. These include the location in the complex plane of singularities of the pointwise Green's function and pinched double roots of the dispersion relation. The primary aim of this work is to establish some rigorous properties related to these criteria and the relationships between them. In the process, we discover that these concepts are not equivalent and point to some interesting consequences for nonlinear front invasion problems. Among the more striking is the fact that pointwise growth does not depend continuously on system parameters. Other results include a determination of the circumstances under which pointwise growth on the real line implies pointwise growth on a semi-infinite interval. As a final application, we consider invasion fronts in an infinite cylinder and show that the linear prediction always favors the formation of stripes in the leading edge.
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Acknowledgments The authors acknowledge partial support through the National Science Foundation (NSF) [DMS-1004517 (M.H.), DMS-0806614 (A.S.), and DMS-1311740 (A.S.)]. This research was initiated during an NSF-sponsored REU program in the summer of 2012 (Bose et al. 2013). We are grateful to Koushiki Bose, Tyler Cox, Stefano Silvestri, and Patrick Varin for working out some of the examples in this article. We also thank Ryan Goh for many stimulating discussions related to the material in Sects. 4 and 5.
- Invasion fronts
- Linear spreading speed
- Pointwise Green's function
- Pointwise growth