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January  2012, 17(1): 127-152. doi: 10.3934/dcdsb.2012.17.127

Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels

Huiling Li 1, , Peter Y. H. Pang 2, and  Mingxin Wang 3,

1. 

Department of Mathematics, Southeast University, Nanjing 210096, China
2. 

Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore
3. 

Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

Received  December 2010 Revised  March 2011 Published  October 2011

In this paper, we consider a mathematical model for a prey-predator dynamical system with diffusion and trophic interactions of three levels. In this model, a general trophic function based on the ratio between the prey and a linear function of the predator is used at each level. At the two limits of this trophic function, one recovers the classical prey-dependent and ratio-dependent predation models, respectively. We offer a complete discussion of the dynamical behavior of the model under the homogeneous Neumann boundary condition (the same behavior is also seen in the absence of diffusion). We also discuss existence, uniqueness, stability and bifurcation of equilibrium behavior corresponding to positive steady state solutions under the homogeneous Dirichlet boundary condition. Finally, we give interpretations of some of these results in the context of different predation models.
Keywords: trophic function, positive steady state solution., Prey-predator model.
Mathematics Subject Classification: Primary: 35J55, 35B25; Secondary: 92C4.
Citation: Huiling Li, Peter Y. H. Pang, Mingxin Wang. Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 127-152. doi: 10.3934/dcdsb.2012.17.127
References:
[1]

P. A. Abrams and L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15 (2000), 337-341. doi: 10.1016/S0169-5347(00)01908-X.

[2]

C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems. Current developments in partial differential equations, Discrete Contin. Dyn. Syst., 8 (2002), 289-302.

[3]

J. Blat and K. J. Brown, Global bifurcation on positive solutions in some systems of elliptic equations, SIAM. J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.

[4]

G. Buffoni, M. P. Cassinari and M. Groppi, Modelling of predator-prey trophic interactions. Part II: Three trophic levels, J. Math. Biol., 54 (2007), 623-644.

[5]

G. Buffoni, M. P. Cassinari, M. Groppi and M. Serluca, Modelling of predator-prey trophic interactions. Part I: Two trophic levels, J. Math. Biol., 50 (2005), 713-732.

[6]

A. Casal, J. C. Eilbede and J. López-Gómez, Existence and uniqueness of coexitence states for a predator-prey model with diffusion, Differential Integral Equations, 7 (1994), 411-439.

[7]

W.-Y. Chen and M.-X. Wang, Positive steady states of a competitor-competitor-mutualist model, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 53-57. doi: 10.1007/s10255-004-0148-0.

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[9]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[10]

E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM. J. Math. Anal., 34 (2002), 292-314. doi: 10.1137/S0036141001387598.

[11]

M. Delgado, J. López-Gómez and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations, 160 (2000), 175-262.

[12]

Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.

[13]

Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440.

[14]

C. F. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math., 47 (1994), 1571-1594. doi: 10.1002/cpa.3160471203.

[15]

T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.

[16]

W. Ko and K. Ryu, Coexistence states of a predator-prey system with non-monotonic functional response, Nonlinear Anal. RWA, 8 (2007), 769-786. doi: 10.1016/j.nonrwa.2006.03.003.

[17]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1.

[18]

J. López-Gómez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory, Nonlinear Anal., 58 (2004), 749-777. doi: 10.1016/j.na.2004.04.011.

[19]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, New York, 1992.

[20]

R. Peng and M. X. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268. doi: 10.1016/j.jmaa.2005.04.033.

[21]

W. H. Ruan and W. Feng, On the fixed point index and multiple steady states of reaction-diffusion systems, Differential Integral Equations, 8 (1995), 371-391.

[22]

K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135.

[23]

Y. M. Svirezhev and D. O. Logofet, "Stability of Biological Communities," Translated from the Russian by Alexey Voinov, "MIR," Moscow, 1983.

[24]

M. X. Wang, "Nonlinear Partial Differential Equations of Parabolic Type," (Chinese), Science Press, Beijing, 1993.

[25]

M. X. Wang and Q. Wu, Positive solutions of a prey-predator model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718. doi: 10.1016/j.jmaa.2008.04.054.

[26]

Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM. J. Math. Anal., 21 (1990), 327-345. doi: 10.1137/0521018.

show all references

References:
[1]

P. A. Abrams and L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15 (2000), 337-341. doi: 10.1016/S0169-5347(00)01908-X.

[2]

C. O. Alves and D. G. de Figueiredo, Nonvariational elliptic systems. Current developments in partial differential equations, Discrete Contin. Dyn. Syst., 8 (2002), 289-302.

[3]

J. Blat and K. J. Brown, Global bifurcation on positive solutions in some systems of elliptic equations, SIAM. J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.

[4]

G. Buffoni, M. P. Cassinari and M. Groppi, Modelling of predator-prey trophic interactions. Part II: Three trophic levels, J. Math. Biol., 54 (2007), 623-644.

[5]

G. Buffoni, M. P. Cassinari, M. Groppi and M. Serluca, Modelling of predator-prey trophic interactions. Part I: Two trophic levels, J. Math. Biol., 50 (2005), 713-732.

[6]

A. Casal, J. C. Eilbede and J. López-Gómez, Existence and uniqueness of coexitence states for a predator-prey model with diffusion, Differential Integral Equations, 7 (1994), 411-439.

[7]

W.-Y. Chen and M.-X. Wang, Positive steady states of a competitor-competitor-mutualist model, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 53-57. doi: 10.1007/s10255-004-0148-0.

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[9]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7.

[10]

E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM. J. Math. Anal., 34 (2002), 292-314. doi: 10.1137/S0036141001387598.

[11]

M. Delgado, J. López-Gómez and A. Suárez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, J. Differential Equations, 160 (2000), 175-262.

[12]

Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.

[13]

Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440.

[14]

C. F. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure Appl. Math., 47 (1994), 1571-1594. doi: 10.1002/cpa.3160471203.

[15]

T. Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.

[16]

W. Ko and K. Ryu, Coexistence states of a predator-prey system with non-monotonic functional response, Nonlinear Anal. RWA, 8 (2007), 769-786. doi: 10.1016/j.nonrwa.2006.03.003.

[17]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1.

[18]

J. López-Gómez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory, Nonlinear Anal., 58 (2004), 749-777. doi: 10.1016/j.na.2004.04.011.

[19]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations," Plenum Press, New York, 1992.

[20]

R. Peng and M. X. Wang, On multiplicity and stability of positive solutions of a diffusive prey-predator model, J. Math. Anal. Appl., 316 (2006), 256-268. doi: 10.1016/j.jmaa.2005.04.033.

[21]

W. H. Ruan and W. Feng, On the fixed point index and multiple steady states of reaction-diffusion systems, Differential Integral Equations, 8 (1995), 371-391.

[22]

K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135.

[23]

Y. M. Svirezhev and D. O. Logofet, "Stability of Biological Communities," Translated from the Russian by Alexey Voinov, "MIR," Moscow, 1983.

[24]

M. X. Wang, "Nonlinear Partial Differential Equations of Parabolic Type," (Chinese), Science Press, Beijing, 1993.

[25]

M. X. Wang and Q. Wu, Positive solutions of a prey-predator model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718. doi: 10.1016/j.jmaa.2008.04.054.

[26]

Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM. J. Math. Anal., 21 (1990), 327-345. doi: 10.1137/0521018.

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