  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).
