1 Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile.
2 Institut Mathematiques de Bordeaux, Universite Victor Segalen Bordeaux 2, 3 ter Place de la Victoire, 33076 Bordeaux, France.
3 CI 2MA & Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Received: 19 November 2012
Revised: 6 March 2014
In this paper, we consider a reaction-diffusion system describing three interacting species in the food chain structure with nonlocal and cross diffusion. We propose a semi-implicit finite volume scheme for this system, we establish existence and uniqueness of the discrete solution, and it is also showed that the discrete solution generated by the given scheme converges to the corresponding weak solution for the model studied. The convergence proof is based on the use of the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L1 compactness argument that mimics the compactness lemma due to Kruzhkov. Finally we give some numerical examples.
Mathematics Subject Classification: 35K57 / 35M10 / 35A05
Key words: Nonlocal and cross diffusion / food chain model / finite volume scheme
VA was partially supported by CONICYT-Chile through FONDECYT postdoctorado No. 3120197, by project Inserción de Capital Humano Avanzado en la Academia No. 79112012, and DIUBB through project 120808 GI/EF.
© EDP Sciences, SMAI, 2015