• Previous Article
    On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis
  • CPAA Home
  • This Issue
  • Next Article
    Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam
September  2011, 10(5): 1463-1478. doi: 10.3934/cpaa.2011.10.1463

Global attractors of reaction-diffusion systems modeling food chain populations with delays

1. 

Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403

2. 

Department of mathematics, North Carolina State University, Raleigh, NC27695, United States

3. 

Department of Math and Stat. UNCW, 601 S. College Road, Wilmington NC 28403

Received  July 2009 Revised  July 2010 Published  April 2011

In this paper, we study a reaction-diffusion system modeling the population dynamics of a four-species food chain with time delays. Under Dirichlet and Neumann boundary conditions, we discuss the existence of a positive global attractor which demonstrates the presence of a positive steady state and the permanence effect in the ecological system. Sufficient conditions on the interaction rates are given to ensure the persistence of all species in the food chain. For the case of Neumann boundary condition, we further obtain the uniqueness of a positive steady state, and in such case the density functions converge uniformly to a constant solution. Numerical simulations of the food-chain models are also given to demonstrate and compare the asymptotic behavior of the time-dependent density functions.
Citation: Wei Feng, C. V. Pao, Xin Lu. Global attractors of reaction-diffusion systems modeling food chain populations with delays. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1463-1478. doi: 10.3934/cpaa.2011.10.1463
References:
[1]

R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity,, Proc. Roy. Soc. Edinburgh, 123A (1993), 533. Google Scholar

[2]

E. N. Dancer, The existence and uniqueness of positive solutions of competing species equations with diffusion,, Trans. Amer. Math. Soc., 326 (1991), 829. Google Scholar

[3]

Wei Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model,, J. Math. Anal. Appl., 179 (1993), 592. Google Scholar

[4]

Wei Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195. Google Scholar

[5]

W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects,, J. Math. Anal. Appl., 206 (1997), 547. Google Scholar

[6]

W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model,, Diff. Integ. Eq.s, 8 (1995), 617. Google Scholar

[7]

W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays,, Nonlinear Analysis, 26 (1996), 171. Google Scholar

[8]

A. Leung, "Systems of Nonlinear Partial Differential Equations,'', Kluwer Publ., (1989). Google Scholar

[9]

X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays,, Numer. Meth. Part. Diff. Eqn.s, 11 (1995), 591. Google Scholar

[10]

X. Lu, Numerical solutions of coupled parabolic systems with time delays,, Differential Equa tions and Nonlinear Mechanics (Orlando, 528 (2001), 201. Google Scholar

[11]

X. Lu and W. Feng, Periodic solution and oscillation in a competition model with diffusion and distributed delay effects,, Nonlinear Analysis, 27 (1996), 699. Google Scholar

[12]

C. V. Pao, "On Nonlinear Parabolic and Elliptic Equations,'', Plenum Press, (1992). Google Scholar

[13]

C. V. Pao, Numerical methods for semilinear parabolic equations,, SIAM J. Numer. Anal., 24 (1987), 24. Google Scholar

[14]

C. V. Pao, Coupled nonlinear parabolic system with time delays,, J. Math. Anal. Appl., 196 (1995), 237. Google Scholar

[15]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751. Google Scholar

[16]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays,, Nonlinear Analysis T.M.A., 48 (2002), 349. Google Scholar

[17]

C. V. Pao, Numerical analysis of coupled system of nonlinear parabolic equations,, SIAM J. Numer Anal., 36 (1999), 394. Google Scholar

[18]

C. V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays,, J. Math. Anal. Appl., 240 (1999), 249. Google Scholar

[19]

M. H. Posey and A. H. Hines, Complex predator-prey interactions within an estuarine benthic community,, Ecology, 72 (1991), 2155. Google Scholar

[20]

J. T. Rowell and Wei Feng, Coexistence and permanence in a four-species food chain model,, Nonlinear Times and Digest, 2 (1995), 191. Google Scholar

show all references

References:
[1]

R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity,, Proc. Roy. Soc. Edinburgh, 123A (1993), 533. Google Scholar

[2]

E. N. Dancer, The existence and uniqueness of positive solutions of competing species equations with diffusion,, Trans. Amer. Math. Soc., 326 (1991), 829. Google Scholar

[3]

Wei Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model,, J. Math. Anal. Appl., 179 (1993), 592. Google Scholar

[4]

Wei Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195. Google Scholar

[5]

W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects,, J. Math. Anal. Appl., 206 (1997), 547. Google Scholar

[6]

W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model,, Diff. Integ. Eq.s, 8 (1995), 617. Google Scholar

[7]

W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays,, Nonlinear Analysis, 26 (1996), 171. Google Scholar

[8]

A. Leung, "Systems of Nonlinear Partial Differential Equations,'', Kluwer Publ., (1989). Google Scholar

[9]

X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays,, Numer. Meth. Part. Diff. Eqn.s, 11 (1995), 591. Google Scholar

[10]

X. Lu, Numerical solutions of coupled parabolic systems with time delays,, Differential Equa tions and Nonlinear Mechanics (Orlando, 528 (2001), 201. Google Scholar

[11]

X. Lu and W. Feng, Periodic solution and oscillation in a competition model with diffusion and distributed delay effects,, Nonlinear Analysis, 27 (1996), 699. Google Scholar

[12]

C. V. Pao, "On Nonlinear Parabolic and Elliptic Equations,'', Plenum Press, (1992). Google Scholar

[13]

C. V. Pao, Numerical methods for semilinear parabolic equations,, SIAM J. Numer. Anal., 24 (1987), 24. Google Scholar

[14]

C. V. Pao, Coupled nonlinear parabolic system with time delays,, J. Math. Anal. Appl., 196 (1995), 237. Google Scholar

[15]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751. Google Scholar

[16]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays,, Nonlinear Analysis T.M.A., 48 (2002), 349. Google Scholar

[17]

C. V. Pao, Numerical analysis of coupled system of nonlinear parabolic equations,, SIAM J. Numer Anal., 36 (1999), 394. Google Scholar

[18]

C. V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays,, J. Math. Anal. Appl., 240 (1999), 249. Google Scholar

[19]

M. H. Posey and A. H. Hines, Complex predator-prey interactions within an estuarine benthic community,, Ecology, 72 (1991), 2155. Google Scholar

[20]

J. T. Rowell and Wei Feng, Coexistence and permanence in a four-species food chain model,, Nonlinear Times and Digest, 2 (1995), 191. Google Scholar

[1]

Jing Liu, Xiaodong Liu, Sining Zheng, Yanping Lin. Positive steady state of a food chain system with diffusion. Conference Publications, 2007, 2007 (Special) : 667-676. doi: 10.3934/proc.2007.2007.667

[2]

Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771

[3]

Elvira Barbera, Giancarlo Consolo, Giovanna Valenti. A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain. Mathematical Biosciences & Engineering, 2015, 12 (3) : 451-472. doi: 10.3934/mbe.2015.12.451

[4]

B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077

[5]

Lijuan Wang, Hongling Jiang, Ying Li. Positive steady state solutions of a plant-pollinator model with diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1805-1819. doi: 10.3934/dcdsb.2015.20.1805

[6]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[7]

Maria Paola Cassinari, Maria Groppi, Claudio Tebaldi. Effects of predation efficiencies on the dynamics of a tritrophic food chain. Mathematical Biosciences & Engineering, 2007, 4 (3) : 431-456. doi: 10.3934/mbe.2007.4.431

[8]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515

[9]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989

[10]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841

[11]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

[12]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179

[13]

Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137

[14]

Guichen Lu, Zhengyi Lu. Permanence for two-species Lotka-Volterra cooperative systems with delays. Mathematical Biosciences & Engineering, 2008, 5 (3) : 477-484. doi: 10.3934/mbe.2008.5.477

[15]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

[16]

Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic & Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253

[17]

Andrei Korobeinikov, William T. Lee. Global asymptotic properties for a Leslie-Gower food chain model. Mathematical Biosciences & Engineering, 2009, 6 (3) : 585-590. doi: 10.3934/mbe.2009.6.585

[18]

Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373

[19]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105

[20]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]