Approximate Green functions as a tool to prove correctness of a formal approximation in a model of competing and diffusing species (Q1820290)

The authors model the spatial distribution of two competing and diffusing species with the coupld pair of differential equations (1.1) \(\epsilon^ 2u''=f(x,u,v)\) \(v''=g(x,u,v)\) and Neumann boundary condition (1.2) \(u'(- 1)=v'(-1)=0\), \(u'(1)=v'(1)=0\). They derive some new results concerning this singular perturbation problem and demonstrate the use of approximate Green's functions to prove the validity of a constructed formal approximation. In particular they prove the validity of the constructed approximation under precisely the same conditions necessary to be able to do the construction. This removes an unnecessary restriction that was present in earlier work.