The uniform spreading speed in cooperative systems with non-uniform initial data

TL;DR

This paper demonstrates that in cooperative nonlocal dispersal systems with non-uniform initial data, all components share a uniform spreading speed determined by the slowest decay rate, influenced by cooperation and irreducibility.

Contribution

It establishes the uniform spreading speed phenomenon in cooperative systems with non-uniform initial data, extending known results from reaction-diffusion equations.

Findings

All components have a uniform spreading speed depending on the smallest decay rate.

The component with the slowest decay rate can accelerate the spread of others.

Methods apply to classical diffusion systems as well.

Abstract

This paper considers the spreading speed of cooperative nonlocal dispersal system with irreducible reaction functions and non-uniform initial data. Here the non-uniformity means that all components of initial data decay exponentially but their decay rates are different. It is well-known that in a monostable reaction-diffusion or nonlocal dispersal equation, different decay rates of initial data yield different spreading speeds. In this paper, we show that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will possess a uniform spreading speed which non-increasingly depends only on the smallest decay rate of initial data. The nonincreasing property of the uniform spreading speed further implies that the component with the smallest decay rate can accelerate the spatial propagation of other components. In addition, all the methods in this paper can be carried over to the cooperative system with classical diffusion (i.e. random diffusion).