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July  2022, 27(7): 3913-3931. doi: 10.3934/dcdsb.2021211

Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms

Willian Cintra , , Carlos Alberto dos Santos and  Jiazheng Zhou

Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília- DF, Brazil

* Corresponding author: Willian Cintra

Received  March 2021 Revised  May 2021 Published  July 2022 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.

Keywords: Quasilinear elliptic system, global bifurcation, coexistence states, sub- and supersolution method, population dynamics.
Mathematics Subject Classification: Primary: 35J57, 35J62; Secondary: 92D25.
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 and Continuous Dynamical Systems - B, 2022, 27 (7) : 3913-3931. 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.

[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.

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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