American Institute of Mathematical Sciences

Advanced
September  2013, 12(5): 2189-2201. doi: 10.3934/cpaa.2013.12.2189

Existence of positive steady states for a predator-prey model with diffusion

Wenshu Zhou 1, , Hongxing Zhao 2, , Xiaodan Wei 3, and  Guokai Xu 4,

1. 

Department of Mathematics, Dalian Nationalities University, Dalian 116600, China
2. 

College of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China
3. 

School of Computer Science, Dalian Nationalities University, Dalian 116600, China
4. 

College of Electromechanical and Information Engineering, Dalian Nationalities University, Dalian 116600, China

Received  June 2012 Revised  August 2012 Published  January 2013

In this paper, we are concerned with the existence of positive steady states for a diffusive predator-prey model in a spatially heterogeneous environment. We completely determine the intervals of certain parameter of the model in which a positive steady state exists.
Keywords: existence, nonexistence., steady state, Prey-predator model.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J6.
Citation: Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189
References:
[1]

R. S. Cantrell and C. Cosner, Practical persistence in ecological models via comparison methods, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 247-272.  Google Scholar

[2]

E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal, 34 (2002), 292-314. doi: 10.1137/S0036141001387598.  Google Scholar

[3]

Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[4]

Y. H. Du and S. J. Li, Positive solutions with prescribed patterns in some simple semilinear equations, Differential Integral Equations, 15 (2002), 805-822.  Google Scholar

[5]

Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc, 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.  Google Scholar

[6]

Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[7]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Tran. Amer. Math. Soc, 359 (2007), 4557-4593. doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[8]

J. M. Fraile, P. K. Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar

[9]

R. T. Gong and S. B. Hsu, Stability analysis for a class of diffusive coupled system with application to population biology, Can. Appl. Math. Quart, 8 (2000), 79-96.  Google Scholar

[10]

K. Hasík, On a predator-prey system of Gause type, J. Math. Biol., 60 (2010), 59-74. doi: 10.1007/s00285-009-0257-8.  Google Scholar

[11]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwannese J. Mathematics, 9 (2005), 151-173.  Google Scholar

[12]

W. Ko, K. Ryu, A qualitative study on general Gause type predator-prey models with constant diffusion rates, J. Math. Anal. Appl., 344 (2008), 217-230. doi: 10.1016/j.jmaa.2008.03.006.  Google Scholar

[13]

W. Ko, K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal. RWA, 10 (2009), 2558-2573. doi: 10.1016/j.nonrwa.2008.05.012.  Google Scholar

[14]

Y. Kuang, Global stability of Gause-type predator-prey systems, J. Math. Biol., 28 (1990), 463-474. doi: 10.1007/BF00178329.  Google Scholar

[15]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  Google Scholar

[16]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[17]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  Google Scholar

[18]

J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989.  Google Scholar

[19]

T. C. Ouyang, On the positive solutions of semilinear equations $\Delta u + \lambda u-hu^p = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.1090/S0002-9947-1992-1055810-7.  Google Scholar

[20]

Peter Y. H. Pang and M. X. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London. Math. Soc., 88 (2004), 135-157. doi: 10.1112/S0024611503014321.  Google Scholar

[21]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, New York, 1994.  Google Scholar

[22]

M. X. Wang, Peter Y. H. Pang and W. Y. Chen, Sharp spatial pattern of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA Journal of Applied Mathematics, 73 (2008), 815-835. doi: 10.1093/imamat/hxn016.  Google Scholar

show all references

References:
[1]

R. S. Cantrell and C. Cosner, Practical persistence in ecological models via comparison methods, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 247-272.  Google Scholar

[2]

E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal, 34 (2002), 292-314. doi: 10.1137/S0036141001387598.  Google Scholar

[3]

Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010.  Google Scholar

[4]

Y. H. Du and S. J. Li, Positive solutions with prescribed patterns in some simple semilinear equations, Differential Integral Equations, 15 (2002), 805-822.  Google Scholar

[5]

Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc, 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4.  Google Scholar

[6]

Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91. doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[7]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Tran. Amer. Math. Soc, 359 (2007), 4557-4593. doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[8]

J. M. Fraile, P. K. Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar

[9]

R. T. Gong and S. B. Hsu, Stability analysis for a class of diffusive coupled system with application to population biology, Can. Appl. Math. Quart, 8 (2000), 79-96.  Google Scholar

[10]

K. Hasík, On a predator-prey system of Gause type, J. Math. Biol., 60 (2010), 59-74. doi: 10.1007/s00285-009-0257-8.  Google Scholar

[11]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwannese J. Mathematics, 9 (2005), 151-173.  Google Scholar

[12]

W. Ko, K. Ryu, A qualitative study on general Gause type predator-prey models with constant diffusion rates, J. Math. Anal. Appl., 344 (2008), 217-230. doi: 10.1016/j.jmaa.2008.03.006.  Google Scholar

[13]

W. Ko, K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal. RWA, 10 (2009), 2558-2573. doi: 10.1016/j.nonrwa.2008.05.012.  Google Scholar

[14]

Y. Kuang, Global stability of Gause-type predator-prey systems, J. Math. Biol., 28 (1990), 463-474. doi: 10.1007/BF00178329.  Google Scholar

[15]

C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  Google Scholar

[16]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[17]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  Google Scholar

[18]

J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989.  Google Scholar

[19]

T. C. Ouyang, On the positive solutions of semilinear equations $\Delta u + \lambda u-hu^p = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.1090/S0002-9947-1992-1055810-7.  Google Scholar

[20]

Peter Y. H. Pang and M. X. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London. Math. Soc., 88 (2004), 135-157. doi: 10.1112/S0024611503014321.  Google Scholar

[21]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, New York, 1994.  Google Scholar

[22]

M. X. Wang, Peter Y. H. Pang and W. Y. Chen, Sharp spatial pattern of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA Journal of Applied Mathematics, 73 (2008), 815-835. doi: 10.1093/imamat/hxn016.  Google Scholar

[1]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[2]

Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173
[3]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[4]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026
[5]

Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737
[6]

Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021136
[7]

R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423
[8]

Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180
[9]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536
[10]

Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061
[11]

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, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875
[12]

J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059
[13]

Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172
[14]

Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048
[15]

Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046
[16]

Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245
[17]

Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021035
[18]

Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064
[19]

Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597
[20]

Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences