American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme
  • DCDS-B Home
  • This Issue
  • Next Article
    Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system
doi: 10.3934/dcdsb.2021211
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & 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

[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.   Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.   Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[2]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017
[3]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[4]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013
[5]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[6]

Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
[7]

Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009
[8]

Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063
[9]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[10]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[11]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[12]

Pablo Álvarez-Caudevilla, Julián López-Gómez. Characterizing the existence of coexistence states in a class of cooperative systems. Conference Publications, 2009, 2009 (Special) : 24-33. doi: 10.3934/proc.2009.2009.24
[13]

King-Yeung Lam, Wei-Ming Ni. Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1051-1067. doi: 10.3934/dcds.2010.28.1051
[14]

Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21
[15]

Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541
[16]

Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335
[17]

Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328
[18]

Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5189-5202. doi: 10.3934/dcds.2013.33.5189
[19]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515
[20]

Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (119)
  • HTML views (114)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences