Numerical analysis for a three interacting species model with nonlocal and cross diffusion
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 171-192.

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 L 1 compactness argument that mimics the compactness lemma due to Kruzhkov. Finally we give some numerical examples.

Reçu le :
DOI : 10.1051/m2an/2014028
Classification : 35K57, 35M10, 35A05
Mots clés : Nonlocal and cross diffusion, food chain model, finite volume scheme
Anaya, Verónica 1 ; Bendahmane, Mostafa 2 ; Sepúlveda, Mauricio 3

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 CI2MA & Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
@article{M2AN_2015__49_1_171_0,
     author = {Anaya, Ver\'onica and Bendahmane, Mostafa and Sep\'ulveda, Mauricio},
     title = {Numerical analysis for a three interacting species model with nonlocal and cross diffusion},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {171--192},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {1},
     year = {2015},
     doi = {10.1051/m2an/2014028},
     mrnumber = {3342196},
     zbl = {1314.65115},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2014028/}
}
TY  - JOUR
AU  - Anaya, Verónica
AU  - Bendahmane, Mostafa
AU  - Sepúlveda, Mauricio
TI  - Numerical analysis for a three interacting species model with nonlocal and cross diffusion
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 171
EP  - 192
VL  - 49
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2014028/
DO  - 10.1051/m2an/2014028
LA  - en
ID  - M2AN_2015__49_1_171_0
ER  - 
%0 Journal Article
%A Anaya, Verónica
%A Bendahmane, Mostafa
%A Sepúlveda, Mauricio
%T Numerical analysis for a three interacting species model with nonlocal and cross diffusion
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 171-192
%V 49
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2014028/
%R 10.1051/m2an/2014028
%G en
%F M2AN_2015__49_1_171_0
Anaya, Verónica; Bendahmane, Mostafa; Sepúlveda, Mauricio. Numerical analysis for a three interacting species model with nonlocal and cross diffusion. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 171-192. doi : 10.1051/m2an/2014028. http://archive.numdam.org/articles/10.1051/m2an/2014028/

B. Ainseba, M. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlin. Anal.: Real World Applications 8 (2008) 2086–2105. | DOI | MR | Zbl

V. Anaya, M. Bendahmane and M. Sepúlveda, Mathematical and numerical analysis for reaction-diffusion systems modeling the spread of early tumors. Boletin de la Sociedad Española de Matemática Aplicada 47 (2009) 55–62. | MR | Zbl

V. Anaya, M. Bendahmane and M. Sepúlveda, A numerical analysis of a reaction-diffusion system modelling the dynamics of growth tumors. Math. Models Methods Appl. Sci. 20 (2010) 731–756. | DOI | MR | Zbl

V. Anaya, M. Bendahmane and M. Sepúlveda, Mathematical and numerical analysis for predator-prey system in a polluted environment. Netw. Heterogen. Media 5 (2010) 813–847. | DOI | MR | Zbl

B. Andreianov, M. Bendahmane and R. Ruiz-Baier, Analysis of a finite volume method for a cross-diffusion model in population dynamics. Math. Models Methods Appl. Sci. 21 (2011) 307–344. | DOI | MR | Zbl

M. Bendahmane, Weak and classical solutions to predator-prey system with crossdiffusion. Nonlin. Anal. 73 (2010) 2489–2503. | DOI | MR | Zbl

M. Bendahmane, K. H. Karlsen and J. M. Urbano, On a two-sidedly degenerate chemotaxis model with volume-filling effect, Math. Models Methods Appl. Sci. 17 (2007) 783–804. | DOI | MR | Zbl

M. Bendahmane, T. Lepoutre, A. Marrocco and B. Perthame, Conservative cross diffusions and pattern formation through relaxation, J. Math. Pure Appl. 92 (2009) 651–667. | DOI | MR | Zbl

M. Bendahmane and M. Sepúlveda, Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 823–853. | MR | Zbl

R. Eymard, Th. Gallouët and R. Herbin. Finite volume methods. In: Handb. Numer. Anal., vol. VII. North-Holland, Amsterdam (2000). | MR | Zbl

G. Galiano, M. L. Garzón and A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (2001) 281–295. | MR | Zbl

G. Galiano, M. L. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model, Numer. Math. 93 (2003) 655–673. | DOI | MR | Zbl

A. Hasting and T. Powell, Chaos in a three-species food chain. Ecology 72 (1991) 896–903. | DOI

A. Klebanoff and A. Hastings, Chaos in three species food chains. J. Math. Biol. 32 (1994) 427–451. | DOI | MR | Zbl

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). | MR | Zbl

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion. J. Differ. Eq. 131 (1996) 79–131. | DOI | MR | Zbl

K. Mccann and P. Yodzis, Bifurcation structure of a three-species food chain model. Theoret. Popul. Biol. 48 (1995) 93–125. | DOI | Zbl

Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959) 116–162. | Numdam | MR | Zbl

P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Eq. 200 (2004) 245–273. | DOI | MR | Zbl

M. L. Rosenzweig, Exploitation in three trophic levels. Am. Nat. 107 (1973) 275–294. | DOI

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species. J. Theoret. Biol. 79 (1979) 83–99. | DOI | MR

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, 3rd edition. North-Holland, Amsterdam, reprinted in the AMS Chelsea series, AMS, Providence (2001). | MR

C. Tian, Z. Lin and M. Pedersen, Instability induced by cross-diffusion in reaction-diffusion systems. Nonlin. Anal.: Real World Applications 11 (2010) 1036–1045. | DOI | MR | Zbl

A. Turing, The chemical basis of morphogenesis. Philos. Trans. R. Soc. Ser. B 237 (1952) 37–72. | MR | Zbl

P. Yodzis and S. Innes, Body size and consumer-resource dynamics. Am. Nat. 139 (1992) 1151–1175. | DOI

Z. Wen and C. Zhong, Non-constant positive steady states for the HP food chain system with cross-diffusions. Math. Comput. Model. 51 (2010) 1026–1036. | DOI | MR | Zbl

Cité par Sources :