Criteria for Pointwise Growth and Their Role in Invasion Processes

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.