  The development of macroscopic descriptions for the joint dynamics and
behavior of large heterogeneous ensembles subject to ecological forces like
dispersal remains a central challenge for mathematicians and biological
scientists alike. Over the past century, specific attention has been directed
to the role played by dispersal in shaping plant communities, or on the
dynamics of marine open-ocean and intertidal systems, or on biological
invasions, or on the spread of disease, to name a few. Mathematicians and
theoreticians, starting with the efforts of researchers that include Aronson,
Fisher, Kolmogorov, Levin, Okubo, Skellam, Slobodkin, Weinberger and many
others, set the foundation of a fertile area of research at the interface of
ecology, mathematics, population biology and evolutionary biology.
Integrodifference systems, the subject of this manuscript, arise naturally in
the study of the spatial dispersal of organisms whose local population dynamics
are prescribed by models with discrete generations. The brunt of the
mathematical research has focused on the the study of existence traveling wave
solutions and characterizations of the spreading speed particularly, in the
context of cooperative systems. In this paper, we characterize the spreading
speed for a large class of non cooperative systems, all formulated in terms of
integrodifference equations, by the convergence of initial data to wave
solutions. In this setting, the spreading speed is characterized as the slowest
speed of a family of non-constant traveling wave solutions. Our results are
applied to a specific non-cooperative competition system in detail.
