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doi: 10.3934/dcdsb.2021211
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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

* Corresponding author: Willian Cintra

Received  March 2021 Revised  May 2021 Early access August 2021

Fund Project: The first author was partially supported by the project CAPES/Brazil - PrInt $ n^o $ $ 88887.466484/2019-00 $. The second author was partially supported by the project CAPES/Brazil - PrInt $ n^o $ $ 88887.466484/2019 - 00 $ and CNPq/Brazil with the grant $ 311562/2020 - 5 $. The third autor is supported by the project CAPES/Brazil Proc. $ n^o $ $ 2788/2015-02 $

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.

Citation: Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021211
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.  Google Scholar

[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.  Google Scholar

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A. CasalJ. 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.   Google Scholar

[4]

Y. S. ChoiR. 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.  Google Scholar

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W. CintraC. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Secondorder, Classics in Mathematics, Springer-Verlag, Berlin, 2001, reprint of the 1998 edition.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. LeL. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8664.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theo. Biology, 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[29]

H. YuanJ. WuY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. CasalJ. 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.   Google Scholar

[4]

Y. S. ChoiR. 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.  Google Scholar

[5]

W. CintraC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. LeL. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8664.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theo. Biology, 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[29]

H. YuanJ. WuY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The region of coexistence of (1): on the left side, we have the general setting; the dark part represents the region of non-existence of coexistence states, and the area with dashed line represents the region of coexistence state. On the right, the portion with dashed lines represents the region of coexistence states in the simplest case $ d \equiv d(0) $, $ P\equiv 1 $ and $ R \equiv 1 $
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