Spreading speeds and traveling waves for non-cooperative integro-difference systems

TL;DR

This paper analyzes the spreading speeds and traveling wave solutions in non-cooperative integrodifference systems, extending classical results from cooperative systems to more complex ecological models with heterogeneous populations.

Contribution

It characterizes the spreading speed for a broad class of non-cooperative integrodifference systems, which was previously mainly studied in cooperative contexts.

Findings

Spreading speed is the slowest wave speed for non-constant traveling wave solutions.

Convergence of initial data to wave solutions determines spreading speed.

Application to a specific non-cooperative competition system demonstrates the theory.

Abstract

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.