Loading [MathJax]/jax/output/SVG/jax.js

Mathematical and numerical analysis for Predator-prey system in a polluted environment

  • Received: 01 January 2010 Revised: 01 April 2010
  • Primary: 35K57, 35M10; Secondary: 35A05.

  • In this paper, we prove existence results for a Predator-prey system in a polluted environment. The existence result is proved by the Schauder fixed-point theorem. Moreover, we construct a combined finite volume - finite element scheme to our model, we establish existence of discrete solutions to this scheme, and show that it converges to a weak solution. The convergence proof is based on deriving series of a priori estimates and using a general Lp compactness criterion. Finally we give some numerical examples.

    Citation: Verónica Anaya, Mostafa Bendahmane, Mauricio Sepúlveda. Mathematical and numerical analysis for Predator-prey system in a polluted environment[J]. Networks and Heterogeneous Media, 2010, 5(4): 813-847. doi: 10.3934/nhm.2010.5.813

    Related Papers:

    [1] Verónica Anaya, Mostafa Bendahmane, Mauricio Sepúlveda . Mathematical and numerical analysis for Predator-prey system in a polluted environment. Networks and Heterogeneous Media, 2010, 5(4): 813-847. doi: 10.3934/nhm.2010.5.813
    [2] Mostafa Bendahmane . Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Networks and Heterogeneous Media, 2008, 3(4): 863-879. doi: 10.3934/nhm.2008.3.863
    [3] Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie . A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11(1): 1-27. doi: 10.3934/nhm.2016.11.1
    [4] Yue Tai, Xiuli Wang, Weishi Yin, Pinchao Meng . Weak Galerkin method for the Navier-Stokes equation with nonlinear damping term. Networks and Heterogeneous Media, 2024, 19(2): 475-499. doi: 10.3934/nhm.2024021
    [5] François James, Nicolas Vauchelet . One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation. Networks and Heterogeneous Media, 2016, 11(1): 163-180. doi: 10.3934/nhm.2016.11.163
    [6] Rinaldo M. Colombo, Francesca Marcellini, Elena Rossi . Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results. Networks and Heterogeneous Media, 2016, 11(1): 49-67. doi: 10.3934/nhm.2016.11.49
    [7] Zhangxin Chen . On the control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media, 2006, 1(4): 689-706. doi: 10.3934/nhm.2006.1.689
    [8] Verónica Anaya, Mostafa Bendahmane, David Mora, Ricardo Ruiz Baier . On a vorticity-based formulation for reaction-diffusion-Brinkman systems. Networks and Heterogeneous Media, 2018, 13(1): 69-94. doi: 10.3934/nhm.2018004
    [9] Jérôme Droniou . Remarks on discretizations of convection terms in Hybrid mimetic mixed methods. Networks and Heterogeneous Media, 2010, 5(3): 545-563. doi: 10.3934/nhm.2010.5.545
    [10] Paola Goatin, Sheila Scialanga . Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks and Heterogeneous Media, 2016, 11(1): 107-121. doi: 10.3934/nhm.2016.11.107
  • In this paper, we prove existence results for a Predator-prey system in a polluted environment. The existence result is proved by the Schauder fixed-point theorem. Moreover, we construct a combined finite volume - finite element scheme to our model, we establish existence of discrete solutions to this scheme, and show that it converges to a weak solution. The convergence proof is based on deriving series of a priori estimates and using a general Lp compactness criterion. Finally we give some numerical examples.


    [1] A. S. Ackleh and L. Ke, Existence-uniqueness and long time behavior for a class on nonlocal nonlinear parabolic evolution equations, Proc. Amer. Math. Soc., 128 (2000), 3483-3492. doi: 10.1090/S0002-9939-00-05912-8
    [2] V. Anaya, M. Bendahmane and M. Sepúlveda, Mathematical and numerical analysis for reaction-diffusion systems modeling the spread of early tumors, Bol. Soc. Esp. Mat. Apl., (2009), 55-62.
    [3] 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: 10.1142/S0218202510004428
    [4] B. Ainseba, M. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 128 (2008), 2086-2105. doi: 10.1016/j.nonrwa.2007.06.017
    [5] L. Bai and K. Wang, A diffusive stage-structured model in a polluted environment, Nonlinear Anal. Real World Appl., 7 (2006), 96-108. doi: 10.1016/j.nonrwa.2004.11.010
    [6] 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: 10.1142/S0218202507002108
    [7] 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. doi: 10.3934/dcdsb.2009.11.823
    [8] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problem, Nonlinear Anal., 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7
    [9] B. Dubey and J. Hussain, Modelling the interaction of two biological species in a polluted environment, J. Math. Anal. Appl., 246 (2000), 58-79. doi: 10.1006/jmaa.2000.6741
    [10] B. Dubey and J. Hussain, Models for the effect of environmental pollution on forestry resources with time delay, Nonlinear Anal. Real World Appl., 5 (2004), 549-570. doi: 10.1016/j.nonrwa.2004.01.001
    [11] R. Eymard, Th. Gallouët and R. Herbin, "Finite Volume Methods. Handbook of Numerical Analysis," vol. VII, North-Holland, Amsterdam, 2000.
    [12] R. Eymard, D. Hilhorst and M. Vohralík, A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems, Numer. Math., 105 (2006), 73-131. doi: 10.1007/s00211-006-0036-z
    [13] H. I. Freedman and J. B. Shukla, Models for the effects of toxicant in single-species and predator-prey systems, J. Math. Biol., 30 (1991), 15-30. doi: 10.1007/BF00168004
    [14] T. G. Hallam, C. E. Clark and R. R. Lassider, Effects of toxicants on populations: A qualitative approach I. Equilibrium environment exposured, Ecol. Model, 18 (1983), 291-304. doi: 10.1016/0304-3800(83)90019-4
    [15] T. G. Hallam, C. E. Clark and G. S Jordan, Effects of toxicants on populations: A qualitative approach II. First order kinetics, J. Math. Biol., 18 (1983), 25-37.
    [16] T. G. Hallam and J. T. De Luna, Effects of toxicants on populations: A qualitative approach III. Environment and food chains pathways, J. Theor. Biol., 109 (1984), 11-29. doi: 10.1016/S0022-5193(84)80090-9
    [17] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, 1969.
    [18] C. A. Raposo, M. Sepúlveda, O. Vera, D. Carvalho Pereira and M. Lima Santos, Solution and asymptotic behavior for a nonlocal coupled system of reaction-diffusion, Acta Appl. Math. 102 (2008), 37-56. doi: 10.1007/s10440-008-9207-5
    [19] J. Simon, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360
    [20] J. B. Shukla and B. Dubey, Simultaneous effect of two toxicants on biological species: A mathematical model, J. Biol. Syst., 4 (1996), 109-130. doi: 10.1142/S0218339096000090
    [21] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," 3rd revised edition, North-Holland, Amsterdam, reprinted in the AMS Chelsea series, AMS, Providence, 2001.
    [22] M. Vohralik, "Numerical Methods for Nonlinear Elliptic and Parabolic Equations. Application to Flow Problems in Porous and Fractured Media," Ph.D. dissertation, Université de Paris-Sud & Czech Technical University, Prague, 2004.
    [23] X. Yang, Z. Jin and Y. Xue, Weak average persistence and extinction of a predator-prey system in a polluted environment with impulsive toxicant input, Chaos Solitons Fractals, 31 (2007), 726-735. doi: 10.1016/j.chaos.2005.10.042
    [24] K. Yosida, "Functional Analysis and its Applications," New York, Springer-Verlag, 1971.
  • This article has been cited by:

    1. Verónica Anaya, Mostafa Bendahmane, Mauricio Sepúlveda, Numerical analysis for a three interacting species model with nonlocal and cross diffusion, 2015, 49, 0764-583X, 171, 10.1051/m2an/2014028
    2. Raimund Bürger, Ricardo Ruiz-Baier, Canrong Tian, Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator–prey model, 2017, 132, 03784754, 28, 10.1016/j.matcom.2016.06.002
    3. J. Manimaran, L. Shangerganesh, Amar Debbouche, Finite element error analysis of a time-fractional nonlocal diffusion equation with the Dirichlet energy, 2021, 382, 03770427, 113066, 10.1016/j.cam.2020.113066
    4. Lili Chang, Zhen Jin, Efficient numerical methods for spatially extended population and epidemic models with time delay, 2018, 316, 00963003, 138, 10.1016/j.amc.2017.08.028
    5. J. Manimaran, L. Shangerganesh, Error estimates for Galerkin finite element approximations of time-fractional nonlocal diffusion equation, 2021, 98, 0020-7160, 1365, 10.1080/00207160.2020.1820492
    6. L. Shangerganesh, N. Barani Balan, K. Balachandran, Weak-renormalized solutions for predator–prey system, 2013, 92, 0003-6811, 441, 10.1080/00036811.2011.625014
    7. Nguyen Huy Tuan, Daniel Lesnic, Phan Thi Khanh Van, Identification of the initial population of a nonlinear predator-prey system backwards in time, 2019, 479, 0022247X, 1195, 10.1016/j.jmaa.2019.06.075
  • Reader Comments
  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4356) PDF downloads(108) Cited by(7)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog