September  2014, 34(9): 3875-3899. doi: 10.3934/dcds.2014.34.3875

Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion

1. 

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, United States

Received  August 2013 Revised  November 2013 Published  March 2014

In this paper we consider a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion under zero Dirichlet boundary condition. By using topological degree theory, bifurcation theory, energy estimates and asymptotic behavior analysis, we prove the existence, uniqueness and multiplicity of positive steady states solutions under certain conditions on the parameters.
Citation: Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
References:
[1]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,, Appl. Math. Lett., 16 (2003), 1069. doi: 10.1016/S0893-9659(03)90096-6.

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons Ltd., (2003). doi: 10.1002/0470871296.

[3]

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

[4]

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

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.

[6]

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

[7]

E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729. doi: 10.1090/S0002-9947-1984-0743741-4.

[8]

E. N. Dancer and Y. Du, Competing species equations with diffusion, large interactions, and jumpting nonlinearities,, J. Differential Equaitons, 114 (1994), 434. doi: 10.1006/jdeq.1994.1156.

[9]

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

[10]

Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model,, J. Differential Equaitons, 144 (1998), 390. doi: 10.1006/jdeq.1997.3394.

[11]

Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model,, J. Differential Equaitons, 246 (2009), 3932. doi: 10.1016/j.jde.2008.11.007.

[12]

Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment,, In Nonlinear dynamics and evolution equations, (2006), 95.

[13]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone,, J. Differential Equaitons, 229 (2006), 63. doi: 10.1016/j.jde.2006.01.013.

[14]

B. Dubey, B. Das and J. Hussain, A predator-prey interaction model with self and cross-diffusion,, Ecological Modelling, 141 (2001), 67. doi: 10.1016/S0304-3800(01)00255-1.

[15]

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

[16]

C. S. Holling, Some characteristics of simple types of predation and parasitism,, The Canadian Entomologist, 91 (1959), 382. doi: 10.4039/Ent91385-7.

[17]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics,, Hiroshima Math. J., 23 (1993), 509.

[18]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics. Springer-Verlag, (1995).

[19]

K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion,, J. Differential Equations, 197 (2004), 293. doi: 10.1016/j.jde.2003.10.016.

[20]

K. Kuto, Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment,, Nonlinear Anal. Real World Appl., 10 (2009), 943. doi: 10.1016/j.nonrwa.2007.11.015.

[21]

K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion,, J. Differential Equations, 197 (2004), 315. doi: 10.1016/j.jde.2003.08.003.

[22]

K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems with large cross-diffusion,, Appl. Anal., 89 (2010), 1037. doi: 10.1080/00036811003627534.

[23]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species,, Biometrika, 47 (1960), 219.

[24]

A. W. Leung, Nonlinear Systems of Partial Differential Equations,, World Scientific Publishing Co. Pte. Ltd., (2009). doi: 10.1142/9789814277709.

[25]

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

[26]

Z. Lin and M. Pedersen, Coexistence of a general elliptic system in population dynamics,, Comput. Math. Appl., 48 (2004), 617. doi: 10.1016/j.camwa.2003.01.016.

[27]

J. López-Gómez, Positive periodic solutions of Lotka-Volterra reaction-diffusion systems,, Differential Integral Equations, 5 (1992), 55.

[28]

J. López-Gómez and R. Pardo, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case,, Differential Integral Equations, 6 (1993), 1025.

[29]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79. doi: 10.1006/jdeq.1996.0157.

[30]

Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach,, J. Differential Equations, 154 (1999), 157. doi: 10.1006/jdeq.1998.3559.

[31]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics,, Hiroshima Math. J., 11 (1981), 621.

[32]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035.

[33]

K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion,, Adv. Differential Equations, 1 (1996), 1099.

[34]

C. Neuhauser, Mathematical challenges in spatial ecology,, Notices Amer. Math. Soc., 48 (2001), 1304.

[35]

K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone,, J. Differential Equations, 250 (2011), 3988. doi: 10.1016/j.jde.2011.01.026.

[36]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Springer-Verlag, (2001).

[37]

C. V. Pao, Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion,, Nonlinear Anal., 60 (2005), 1197. doi: 10.1016/j.na.2004.10.008.

[38]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Functional Analysis, 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9.

[39]

W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients,, J. Math. Anal. Appl., 197 (1996), 558. doi: 10.1006/jmaa.1996.0039.

[40]

K. Ryu and I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures,, Bull. Korean Math. Soc., 38 (2001), 643.

[41]

K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics,, J. Math. Anal. Appl., 283 (2003), 46. doi: 10.1016/S0022-247X(03)00162-8.

[42]

K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions,, Discrete Contin. Dyn. Syst., 9 (2003), 1049. doi: 10.3934/dcds.2003.9.1049.

[43]

J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. doi: 10.1006/jfan.1999.3483.

[44]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009.

[45]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3.

[46]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion,, In Handbook of differential equations: stationary partial differential equations. Vol. VI, (2008), 411. doi: 10.1016/S1874-5733(08)80023-X.

[47]

C.-H. Zhang and X. P. Yan, Positive solutions bifurcating from zero solution in a Lotka-Volterra competitive system with cross-diffusion effects,, Appl. Math. J. Chinese Univ. Ser. B, 26 (2011), 342. doi: 10.1007/s11766-011-2737-z.

[48]

J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes,, J. Math. Anal. Appl., 389 (2012), 1380. doi: 10.1016/j.jmaa.2012.01.013.

[49]

J. Zhou and J. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses,, J. Math. Anal. Appl., 405 (2013), 618. doi: 10.1016/j.jmaa.2013.03.064.

show all references

References:
[1]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,, Appl. Math. Lett., 16 (2003), 1069. doi: 10.1016/S0893-9659(03)90096-6.

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons Ltd., (2003). doi: 10.1002/0470871296.

[3]

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

[4]

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

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.

[6]

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

[7]

E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Amer. Math. Soc., 284 (1984), 729. doi: 10.1090/S0002-9947-1984-0743741-4.

[8]

E. N. Dancer and Y. Du, Competing species equations with diffusion, large interactions, and jumpting nonlinearities,, J. Differential Equaitons, 114 (1994), 434. doi: 10.1006/jdeq.1994.1156.

[9]

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

[10]

Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model,, J. Differential Equaitons, 144 (1998), 390. doi: 10.1006/jdeq.1997.3394.

[11]

Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model,, J. Differential Equaitons, 246 (2009), 3932. doi: 10.1016/j.jde.2008.11.007.

[12]

Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment,, In Nonlinear dynamics and evolution equations, (2006), 95.

[13]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone,, J. Differential Equaitons, 229 (2006), 63. doi: 10.1016/j.jde.2006.01.013.

[14]

B. Dubey, B. Das and J. Hussain, A predator-prey interaction model with self and cross-diffusion,, Ecological Modelling, 141 (2001), 67. doi: 10.1016/S0304-3800(01)00255-1.

[15]

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

[16]

C. S. Holling, Some characteristics of simple types of predation and parasitism,, The Canadian Entomologist, 91 (1959), 382. doi: 10.4039/Ent91385-7.

[17]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics,, Hiroshima Math. J., 23 (1993), 509.

[18]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics. Springer-Verlag, (1995).

[19]

K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion,, J. Differential Equations, 197 (2004), 293. doi: 10.1016/j.jde.2003.10.016.

[20]

K. Kuto, Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment,, Nonlinear Anal. Real World Appl., 10 (2009), 943. doi: 10.1016/j.nonrwa.2007.11.015.

[21]

K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion,, J. Differential Equations, 197 (2004), 315. doi: 10.1016/j.jde.2003.08.003.

[22]

K. Kuto and Y. Yamada, Positive solutions for Lotka-Volterra competition systems with large cross-diffusion,, Appl. Anal., 89 (2010), 1037. doi: 10.1080/00036811003627534.

[23]

P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species,, Biometrika, 47 (1960), 219.

[24]

A. W. Leung, Nonlinear Systems of Partial Differential Equations,, World Scientific Publishing Co. Pte. Ltd., (2009). doi: 10.1142/9789814277709.

[25]

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

[26]

Z. Lin and M. Pedersen, Coexistence of a general elliptic system in population dynamics,, Comput. Math. Appl., 48 (2004), 617. doi: 10.1016/j.camwa.2003.01.016.

[27]

J. López-Gómez, Positive periodic solutions of Lotka-Volterra reaction-diffusion systems,, Differential Integral Equations, 5 (1992), 55.

[28]

J. López-Gómez and R. Pardo, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case,, Differential Integral Equations, 6 (1993), 1025.

[29]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79. doi: 10.1006/jdeq.1996.0157.

[30]

Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach,, J. Differential Equations, 154 (1999), 157. doi: 10.1006/jdeq.1998.3559.

[31]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics,, Hiroshima Math. J., 11 (1981), 621.

[32]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035.

[33]

K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion,, Adv. Differential Equations, 1 (1996), 1099.

[34]

C. Neuhauser, Mathematical challenges in spatial ecology,, Notices Amer. Math. Soc., 48 (2001), 1304.

[35]

K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone,, J. Differential Equations, 250 (2011), 3988. doi: 10.1016/j.jde.2011.01.026.

[36]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Springer-Verlag, (2001).

[37]

C. V. Pao, Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion,, Nonlinear Anal., 60 (2005), 1197. doi: 10.1016/j.na.2004.10.008.

[38]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Functional Analysis, 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9.

[39]

W. H. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients,, J. Math. Anal. Appl., 197 (1996), 558. doi: 10.1006/jmaa.1996.0039.

[40]

K. Ryu and I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures,, Bull. Korean Math. Soc., 38 (2001), 643.

[41]

K. Ryu and I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics,, J. Math. Anal. Appl., 283 (2003), 46. doi: 10.1016/S0022-247X(03)00162-8.

[42]

K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions,, Discrete Contin. Dyn. Syst., 9 (2003), 1049. doi: 10.3934/dcds.2003.9.1049.

[43]

J. Shi, Persistence and bifurcation of degenerate solutions,, J. Funct. Anal., 169 (1999), 494. doi: 10.1006/jfan.1999.3483.

[44]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009.

[45]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3.

[46]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion,, In Handbook of differential equations: stationary partial differential equations. Vol. VI, (2008), 411. doi: 10.1016/S1874-5733(08)80023-X.

[47]

C.-H. Zhang and X. P. Yan, Positive solutions bifurcating from zero solution in a Lotka-Volterra competitive system with cross-diffusion effects,, Appl. Math. J. Chinese Univ. Ser. B, 26 (2011), 342. doi: 10.1007/s11766-011-2737-z.

[48]

J. Zhou, Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes,, J. Math. Anal. Appl., 389 (2012), 1380. doi: 10.1016/j.jmaa.2012.01.013.

[49]

J. Zhou and J. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses,, J. Math. Anal. Appl., 405 (2013), 618. doi: 10.1016/j.jmaa.2013.03.064.

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