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On an upper bound for the spreading speed
Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília- DF, Brazil |
In this paper, we present results about existence and non-existence of coexistence states for a reaction-diffusion predator-prey model with the two species living in a bounded region with inhospitable boundary and Holling type II functional response. The predator is a specialist and presents self-diffusion and cross-diffusion behavior. We show the existence of coexistence states by combining global bifurcation theory with the method of sub- and supersolutions.
References:
[1] |
H. Amann,
On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 125-146.
doi: 10.1512/iumj.1971.21.21012. |
[2] |
R. S. Cantrell and C. Cosner,
Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064.
doi: 10.1137/0522068. |
[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), 411-439.
|
[4] |
Y. S. Choi, R. Lui and Y. Yamada,
Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.
doi: 10.3934/dcds.2004.10.719. |
[5] |
W. Cintra, C. Morales-Rodrigo and A. Suárez,
Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations, 267 (2019), 619-657.
doi: 10.1016/j.jde.2019.01.021. |
[6] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[7] |
C. Cosner and A. C. Lazer,
Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132.
doi: 10.1137/0144080. |
[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] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[10] |
Y. Du,
Effects of a degeneracy in the competition model. I. Classical and generalized steady-state solutions, J. Differential Equations, 181 (2002), 92-132.
doi: 10.1006/jdeq.2001.4074. |
[11] |
Y. 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.
doi: 10.1006/jdeq.1997.3394. |
[12] |
L. Dung and H. L. Smith,
Steady states of models of microbial growth and competition with chemotaxis, J. Math. Anal. Appl., 229 (1999), 295-318.
doi: 10.1006/jmaa.1998.6167. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Secondorder, Classics in Mathematics, Springer-Verlag, Berlin, 2001, reprint of the 1998 edition. |
[14] |
W. Ko and K. Ryu,
Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[15] |
K. Kuto and Y. Yamada,
Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.
doi: 10.1016/j.jde.2003.08.003. |
[16] |
K. Kuto and Y. Yamada,
On limit systems for some population models with cross-diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2745-2769.
doi: 10.3934/dcdsb.2012.17.2745. |
[17] |
D. Le, L. V. Nguyen and T. T. Nguyen,
Regularity and coexistence problems for strongly coupled elliptic systems, Indiana Univ. Math. J., 56 (2007), 1749-1791.
doi: 10.1512/iumj.2007.56.2979. |
[18] |
C. Li,
On global bifurcation for a cross-diffusion predator-prey system with prey-taxis, Comput. Math. Appl., 76 (2018), 1014-1025.
doi: 10.1016/j.camwa.2018.05.037. |
[19] |
J. López-Gómez,
Nonlinear eigenvalues and global bifurcation application to the search of positive solutions for general Lotka-Volterra reaction diffusion systems with two species, Differential Integral Equations, 7 (1994), 1427-1452.
|
[20] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8664. |
[21] |
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), 1025-1031.
|
[22] |
Y. Lou and W.-M. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[23] |
K. Nakashima and Y. Yamada,
Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122.
|
[24] |
C. V. Pao,
Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Anal., 6 (2005), 1197-1217.
doi: 10.1016/j.na.2004.10.008. |
[25] |
R. Peng and J. Shi,
Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886.
doi: 10.1016/j.jde.2009.03.008. |
[26] |
W. H. Ruan,
Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl, 197 (1996), 558-578.
doi: 10.1006/jmaa.1996.0039. |
[27] |
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), 46-65.
doi: 10.1016/S0022-247X(03)00162-8. |
[28] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Spatial segregation of interacting species, J. Theo. Biology, 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[29] |
H. Yuan, J. Wu, Y. Jia and H. Nie,
Coexistence states of a predator-prey model with cross-diffusion, Nonlinear Anal. Real World Appl., 41 (2018), 179-203.
doi: 10.1016/j.nonrwa.2017.10.009. |
[30] |
J. Zhou and C. Mu,
Coexistence states of a Holling type-II predator-prey system, J. Math. Anal. Appl, 369 (2010), 555-563.
doi: 10.1016/j.jmaa.2010.04.001. |
[31] |
J. Zhou and C. Mu,
Corrigendum to "Coexistence states of a Holling type-II predator-prey system" [J. Math. Anal. Appl. 369 (2) (2010) 555–563], [mr2651701], J. Math. Anal. Appl., 383 (2011), 636-639.
doi: 10.1016/j.jmaa.2011.06.033. |
show all references
References:
[1] |
H. Amann,
On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 125-146.
doi: 10.1512/iumj.1971.21.21012. |
[2] |
R. S. Cantrell and C. Cosner,
Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064.
doi: 10.1137/0522068. |
[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), 411-439.
|
[4] |
Y. S. Choi, R. Lui and Y. Yamada,
Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.
doi: 10.3934/dcds.2004.10.719. |
[5] |
W. Cintra, C. Morales-Rodrigo and A. Suárez,
Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations, 267 (2019), 619-657.
doi: 10.1016/j.jde.2019.01.021. |
[6] |
C. Cosner,
Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.
doi: 10.3934/dcds.2014.34.1701. |
[7] |
C. Cosner and A. C. Lazer,
Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132.
doi: 10.1137/0144080. |
[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] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[10] |
Y. Du,
Effects of a degeneracy in the competition model. I. Classical and generalized steady-state solutions, J. Differential Equations, 181 (2002), 92-132.
doi: 10.1006/jdeq.2001.4074. |
[11] |
Y. 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.
doi: 10.1006/jdeq.1997.3394. |
[12] |
L. Dung and H. L. Smith,
Steady states of models of microbial growth and competition with chemotaxis, J. Math. Anal. Appl., 229 (1999), 295-318.
doi: 10.1006/jmaa.1998.6167. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Secondorder, Classics in Mathematics, Springer-Verlag, Berlin, 2001, reprint of the 1998 edition. |
[14] |
W. Ko and K. Ryu,
Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[15] |
K. Kuto and Y. Yamada,
Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.
doi: 10.1016/j.jde.2003.08.003. |
[16] |
K. Kuto and Y. Yamada,
On limit systems for some population models with cross-diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2745-2769.
doi: 10.3934/dcdsb.2012.17.2745. |
[17] |
D. Le, L. V. Nguyen and T. T. Nguyen,
Regularity and coexistence problems for strongly coupled elliptic systems, Indiana Univ. Math. J., 56 (2007), 1749-1791.
doi: 10.1512/iumj.2007.56.2979. |
[18] |
C. Li,
On global bifurcation for a cross-diffusion predator-prey system with prey-taxis, Comput. Math. Appl., 76 (2018), 1014-1025.
doi: 10.1016/j.camwa.2018.05.037. |
[19] |
J. López-Gómez,
Nonlinear eigenvalues and global bifurcation application to the search of positive solutions for general Lotka-Volterra reaction diffusion systems with two species, Differential Integral Equations, 7 (1994), 1427-1452.
|
[20] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8664. |
[21] |
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), 1025-1031.
|
[22] |
Y. Lou and W.-M. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[23] |
K. Nakashima and Y. Yamada,
Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122.
|
[24] |
C. V. Pao,
Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Anal., 6 (2005), 1197-1217.
doi: 10.1016/j.na.2004.10.008. |
[25] |
R. Peng and J. Shi,
Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886.
doi: 10.1016/j.jde.2009.03.008. |
[26] |
W. H. Ruan,
Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl, 197 (1996), 558-578.
doi: 10.1006/jmaa.1996.0039. |
[27] |
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), 46-65.
doi: 10.1016/S0022-247X(03)00162-8. |
[28] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Spatial segregation of interacting species, J. Theo. Biology, 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[29] |
H. Yuan, J. Wu, Y. Jia and H. Nie,
Coexistence states of a predator-prey model with cross-diffusion, Nonlinear Anal. Real World Appl., 41 (2018), 179-203.
doi: 10.1016/j.nonrwa.2017.10.009. |
[30] |
J. Zhou and C. Mu,
Coexistence states of a Holling type-II predator-prey system, J. Math. Anal. Appl, 369 (2010), 555-563.
doi: 10.1016/j.jmaa.2010.04.001. |
[31] |
J. Zhou and C. Mu,
Corrigendum to "Coexistence states of a Holling type-II predator-prey system" [J. Math. Anal. Appl. 369 (2) (2010) 555–563], [mr2651701], J. Math. Anal. Appl., 383 (2011), 636-639.
doi: 10.1016/j.jmaa.2011.06.033. |

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