Study On The Approximation Relationship Between The Non-local Interaction System And The Local Reaction-diffusion System

Turing instability is well-known instability caused by diffusion,explaining many pattern formations in the nature.In recent years,non-local interactions have attracted more and more attentions in various fields,also describing a variety of pattern formations.The relationship between the two different systems is the main problem studied in this paper.we consider the following non-local interaction system with Neumann boundary conditions and this following reaction-diffusion system in the following sense: First,the non-local interaction system which is approximated by the local reaction diffusion system under the Neumann boundary condition is found,that is,the form of the non-local interaction operator D,and the approximation relation between the two systems is strictly verified by the energy method,the uniform Gronwall lemma and the positive definite set theory;Second,as for the non-local interaction system under Neumann boundary condition we found,we analyze its eigenvalue at the steady constant solution,thus obtain the instability caused by nonlocal interaction.