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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.

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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