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 compactness argument that mimics the compactness lemma due to Kruzhkov. Finally we give some numerical examples.
DOI : 10.1051/m2an/2014028
Mots clés : Nonlocal and cross diffusion, food chain model, finite volume scheme
@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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an/2014028/
A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlin. Anal.: Real World Applications 8 (2008) 2086–2105. | DOI | MR | Zbl
, and ,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
, and ,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
, and ,Mathematical and numerical analysis for predator-prey system in a polluted environment. Netw. Heterogen. Media 5 (2010) 813–847. | DOI | MR | Zbl
, and ,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
, and ,Weak and classical solutions to predator-prey system with crossdiffusion. Nonlin. Anal. 73 (2010) 2489–2503. | DOI | MR | Zbl
,On a two-sidedly degenerate chemotaxis model with volume-filling effect, Math. Models Methods Appl. Sci. 17 (2007) 783–804. | DOI | MR | Zbl
, and ,Conservative cross diffusions and pattern formation through relaxation, J. Math. Pure Appl. 92 (2009) 651–667. | DOI | MR | Zbl
, , and ,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
and ,R. Eymard, Th. Gallouët and R. Herbin. Finite volume methods. In: Handb. Numer. Anal., vol. VII. North-Holland, Amsterdam (2000). | MR | Zbl
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
, and ,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
, and ,Chaos in a three-species food chain. Ecology 72 (1991) 896–903. | DOI
and ,Chaos in three species food chains. J. Math. Biol. 32 (1994) 427–451. | DOI | MR | Zbl
and ,J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). | MR | Zbl
Diffusion, self-diffusion and cross-diffusion. J. Differ. Eq. 131 (1996) 79–131. | DOI | MR | Zbl
and ,Bifurcation structure of a three-species food chain model. Theoret. Popul. Biol. 48 (1995) 93–125. | DOI | Zbl
and ,On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959) 116–162. | Numdam | MR | Zbl
,Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Eq. 200 (2004) 245–273. | DOI | MR | Zbl
and ,Exploitation in three trophic levels. Am. Nat. 107 (1973) 275–294. | DOI
,Spatial segregation of interacting species. J. Theoret. Biol. 79 (1979) 83–99. | DOI | MR
, and ,R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, 3rd edition. North-Holland, Amsterdam, reprinted in the AMS Chelsea series, AMS, Providence (2001). | MR
Instability induced by cross-diffusion in reaction-diffusion systems. Nonlin. Anal.: Real World Applications 11 (2010) 1036–1045. | DOI | MR | Zbl
, and ,The chemical basis of morphogenesis. Philos. Trans. R. Soc. Ser. B 237 (1952) 37–72. | MR | Zbl
,Body size and consumer-resource dynamics. Am. Nat. 139 (1992) 1151–1175. | DOI
and ,Non-constant positive steady states for the HP food chain system with cross-diffusions. Math. Comput. Model. 51 (2010) 1026–1036. | DOI | MR | Zbl
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