# American Institute of Mathematical Sciences

December  2010, 5(4): 813-847. doi: 10.3934/nhm.2010.5.813

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

 1 Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile 2 Institut de Mathématiques de Bordeaux, Université Victor Segalen Bordeaux 2, 33076 Bordeaux, France 3 CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción

Received  January 2010 Revised  April 2010 Published  November 2010

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 $L^p$ 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. Networks and Heterogeneous Media, 2010, 5 (4) : 813-847. doi: 10.3934/nhm.2010.5.813
##### References:
 [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 $L^p(0,T;B)$, 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.

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##### References:
 [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 $L^p(0,T;B)$, 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.
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