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March  2018, 17(2): 477-485. doi: 10.3934/cpaa.2018026

Nonexistence of nonconstant positive steady states of a diffusive predator-prey model

Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, 264209, China

Received  January 2017 Revised  September 2017 Published  March 2018

Fund Project: The authors are supported by the National Natural Science Foundation of China (No.11771109).

In this paper, we investigate a diffusive predator-prey model with a general predator functional response. We show that there exist no nonconstant positive steady states when the interaction between the predator and prey is strong. This result implies that the global bifurcating branches of steady state solutions are bounded loops for a predator-prey model with Holling type Ⅲ functional response.

Citation: 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
References:
[1]

R. S. Cantrell and C. Cosner, A mathematical model for the propagation of a hantavirus in structured populations, J. Math. Anal. Appl., 257 (2001), 206-222.   Google Scholar

[2]

K. Chaudhuri, Dynamic optimization of combined harvesting of a two species fishery, Ecol. Model., 41 (1988), 17-25.   Google Scholar

[3]

S. Chen and J. Yu, Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator, J. Differential Equations, 260 (2016), 7923-7939.   Google Scholar

[4]

K.-S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.   Google Scholar

[5]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.   Google Scholar

[6]

Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.   Google Scholar

[7]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.   Google Scholar

[8]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.   Google Scholar

[9]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.   Google Scholar

[10]

C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 45 (1965), 1-60.   Google Scholar

[11]

S.-B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911.   Google Scholar

[12]

Y. Kuang and H. I. Freedman, Relaxation oscillation profile of limit cycle in predator-prey system, Math. Biosci., 88 (1988), 67-84.   Google Scholar

[13]

Y. Li and J. Wang, Spatiotemporal patterns of a predator-prey system with an Allee effect and Holling type Ⅲ functional response, Int. J. Bifurcat. Chaos, 26 (2016), 1650088.  Google Scholar

[14]

G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary p in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.   Google Scholar

[15]

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

[16]

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

[17]

Y. LouW.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Cont. Dyn. Syst., 35 (2015), 1589-1607.   Google Scholar

[18]

P. Y. H. Pang and M. 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.   Google Scholar

[19]

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.   Google Scholar

[20]

R. PengJ. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.   Google Scholar

[21]

R. Peng and M. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.   Google Scholar

[22]

M. L. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Amer. Natur., 97 (1963), 209-223.   Google Scholar

[23]

H.-B. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA. J. Appl. Math., 80 (2015), 1534-1568.   Google Scholar

[24]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.   Google Scholar

[25]

P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, Princeton, 2003.  Google Scholar

[26]

A-Y. WanZ.-q. Song and L.-f. Zhang, Patterned solutions of a homogenous diffusive predator-prey system of Holling type-Ⅲ, Acta Math. Appl. Sin. Engl. Ser., 4 (2016), 1073-1086.   Google Scholar

[27]

J. Wang, Spatiotemporal patterns of a homogeneous diffusive predator-prey system with Holling type Ⅲ functional response, To appear in J. Dyn. Diff. Equat. doi: 10.1007/s10884-016-9517-7.  Google Scholar

[28]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.   Google Scholar

[29]

J. WangJ. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.   Google Scholar

[30]

M. WeiJ. Wu and G. Guo, The effect of predator competition on positive solutions for a predator-prey model with diffusion, Nonlinear Anal., 75 (2012), 5053-5068.   Google Scholar

[31]

W.-b. YangJ.-H. Wu and H. Nie, Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14 (2015), 1183-1204.   Google Scholar

[32]

R. Yang and J. Wei, Stability and bifurcation analysis of a diffusive prey-predator system in Holling type Ⅲ with a prey refuge, Nonlinear Dyn., 79 (2015), 631-646.   Google Scholar

[33]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.   Google Scholar

[34]

J. Zhou, Qualitative analysis of a modified Leslie-Gower predator-prey model with Growley-Martin funtional responses, Commun. Pure Appl. Anal., 14 (2015), 1127-1145.   Google Scholar

show all references

References:
[1]

R. S. Cantrell and C. Cosner, A mathematical model for the propagation of a hantavirus in structured populations, J. Math. Anal. Appl., 257 (2001), 206-222.   Google Scholar

[2]

K. Chaudhuri, Dynamic optimization of combined harvesting of a two species fishery, Ecol. Model., 41 (1988), 17-25.   Google Scholar

[3]

S. Chen and J. Yu, Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator, J. Differential Equations, 260 (2016), 7923-7939.   Google Scholar

[4]

K.-S. Cheng, Uniqueness of a limit cycle for a predator-prey system, SIAM J. Math. Anal., 12 (1981), 541-548.   Google Scholar

[5]

Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.   Google Scholar

[6]

Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.   Google Scholar

[7]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.   Google Scholar

[8]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.   Google Scholar

[9]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398.   Google Scholar

[10]

C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 45 (1965), 1-60.   Google Scholar

[11]

S.-B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911.   Google Scholar

[12]

Y. Kuang and H. I. Freedman, Relaxation oscillation profile of limit cycle in predator-prey system, Math. Biosci., 88 (1988), 67-84.   Google Scholar

[13]

Y. Li and J. Wang, Spatiotemporal patterns of a predator-prey system with an Allee effect and Holling type Ⅲ functional response, Int. J. Bifurcat. Chaos, 26 (2016), 1650088.  Google Scholar

[14]

G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary p in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.   Google Scholar

[15]

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

[16]

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

[17]

Y. LouW.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Cont. Dyn. Syst., 35 (2015), 1589-1607.   Google Scholar

[18]

P. Y. H. Pang and M. 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.   Google Scholar

[19]

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.   Google Scholar

[20]

R. PengJ. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.   Google Scholar

[21]

R. Peng and M. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.   Google Scholar

[22]

M. L. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Amer. Natur., 97 (1963), 209-223.   Google Scholar

[23]

H.-B. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA. J. Appl. Math., 80 (2015), 1534-1568.   Google Scholar

[24]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.   Google Scholar

[25]

P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, Princeton, 2003.  Google Scholar

[26]

A-Y. WanZ.-q. Song and L.-f. Zhang, Patterned solutions of a homogenous diffusive predator-prey system of Holling type-Ⅲ, Acta Math. Appl. Sin. Engl. Ser., 4 (2016), 1073-1086.   Google Scholar

[27]

J. Wang, Spatiotemporal patterns of a homogeneous diffusive predator-prey system with Holling type Ⅲ functional response, To appear in J. Dyn. Diff. Equat. doi: 10.1007/s10884-016-9517-7.  Google Scholar

[28]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.   Google Scholar

[29]

J. WangJ. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.   Google Scholar

[30]

M. WeiJ. Wu and G. Guo, The effect of predator competition on positive solutions for a predator-prey model with diffusion, Nonlinear Anal., 75 (2012), 5053-5068.   Google Scholar

[31]

W.-b. YangJ.-H. Wu and H. Nie, Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14 (2015), 1183-1204.   Google Scholar

[32]

R. Yang and J. Wei, Stability and bifurcation analysis of a diffusive prey-predator system in Holling type Ⅲ with a prey refuge, Nonlinear Dyn., 79 (2015), 631-646.   Google Scholar

[33]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.   Google Scholar

[34]

J. Zhou, Qualitative analysis of a modified Leslie-Gower predator-prey model with Growley-Martin funtional responses, Commun. Pure Appl. Anal., 14 (2015), 1127-1145.   Google Scholar

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