Front-like entire solutions for monostable reaction-diffusion systems

TL;DR

This paper investigates the existence and properties of entire solutions for monostable reaction-diffusion systems, including cooperative and non-cooperative cases, with applications to biological and epidemiological models.

Contribution

It introduces novel methods for constructing entire solutions in non-cooperative systems using auxiliary cooperative systems, a first in this research area.

Findings

Existence of entire solutions in cooperative systems established.

Comparison principles used for non-cooperative systems.

Applications demonstrated in biological and epidemiological models.

Abstract

This paper is concerned with front-like entire solutions for monostable reactiondiffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Combining a SIS and traveling fronts with different wave speeds and directions, the existence and various qualitative properties of entire solutions are then established using comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish some comparison arguments for the three systems. The existence of entire solutions is then proved via the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.