Accelerating or not in the spatial propagation of nonlocal dispersal cooperative reducible systems

Abstract

This paper investigates the spatial propagation problem in cooperative recursive systems in the absence of irreducibility, which is a critical assumption to guarantee uniform spatial propagation across all components. When the linearization at zero vector of the reducible system is in Frobenius form, we demonstrate that the i-th component exhibiting accelerated propagation could accelerate the spatial propagation of all other components and the spreading speeds of all components are infinite, provided that the $(i,i)$-entry in the Frobenius matrix belongs to the first diagonal block. This result reveals that uniform propagation of all components can occur even when the irreducibility condition is not satisfied. However, when the $(i,i)$-entry is not in the first diagonal block, some components have finite spreading speeds while others have infinite ones, which implies that the propagation of the system is non-uniform. Moreover, we extend our analysis to nonlocal dispersal cooperative systems and explore a special case where the dispersal kernel of a component has an algebraically decaying tail.