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

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.
Citation: Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[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.

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