doi: 10.3934/dcdsb.2021255
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Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

2. 

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Bin Liu

Received  May 2021 Revised  August 2021 Early access October 2021

Fund Project: This work is supported by National Natural Science Foundation of China grant 11971185

In this paper, we study the prey-predator model with indirect pursuit-evasion interaction defined on a smooth bounded domain with homogeneous Neumann boundary conditions. We obtain the globa existence and boundedness of the classical solution of the model by estimating $ L^{p} $-norm of $ u $ and $ v $, and we also show the large time behavior and convergence rate of the solution.

Citation: Chao Liu, Bin Liu. Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021255
References:
[1]

I. Ahn and C. Yoon, Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis, J. Differential Equations, 268 (2020), 4222-4255.  doi: 10.1016/j.jde.2019.10.019.  Google Scholar

[2]

I. Ahn and C. Yoon, Global solvability of prey–predator models with indirect predator-taxis, Z. Angew. Math. Phys, 72 (2021), Paper No. 29, 20 pp. doi: 10.1007/s00033-020-01461-y.  Google Scholar

[3]

B. E. AinsebaM. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.  Google Scholar

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P. Amorim and B. Telch, A chemotaxis predator-prey model with indirect pursuit-evasion dynamics and parabolic signal, J. Math. Anal. Appl., 500 (2021), Paper No. 125128, 27 pp. doi: 10.1016/j.jmaa.2021.125128.  Google Scholar

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X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

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H. Bréezis and W. A. Strauss, Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

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T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity., 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

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X. He and S. Zheng, Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77.  doi: 10.1016/j.aml.2015.04.017.  Google Scholar

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C. D. HoeflerM. Taylor and E. M. Jakob, Chemosensory response to prey in pkidippus audax (araneae, salticidae) and pardosa milvina (araneae, lycosidae), J. Archnol., 30 (2002), 155-158.   Google Scholar

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D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

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H.-Y. Jin and Z.-A. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.  Google Scholar

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J. M. LeeT. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn, 3 (2009), 551-573.  doi: 10.1080/17513750802716112.  Google Scholar

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G. LiY. Tao and M. Winkler, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst.-Ser. B., 25 (2020), 4383-4396.  doi: 10.3934/dcdsb.2020102.  Google Scholar

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W. W. MurdochJ. Chesson and P. L. Chesson, Biological control in theory and practice, Amer. Nat., 125 (1985), 344-366.   Google Scholar

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G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 545-582.  doi: 10.1016/j.nonrwa.2018.09.020.  Google Scholar

[21]

N. SapoukhinaY. Tyutyunov and R. Arditi, The role of prey taxis in biological control: A spatial theoretical model, Amer. Nat., 162 (2003), 61-76.  doi: 10.1086/375297.  Google Scholar

[22]

Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.  Google Scholar

[23]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[24]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[25]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst.-Ser. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar

[26]

J. Wang and M. Wang, The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741-6762.  doi: 10.1002/mma.5189.  Google Scholar

[27]

J. Wang and M. Wang, The dynamics of a predator-prey model with diffusion and indirect prey-taxis, J. Dyn. Diff. Equat., 32 (2020), 1291-1310.  doi: 10.1007/s10884-019-09778-7.  Google Scholar

[28]

Q. WangY. Song and L. Shao, Nonconstant positive steady states and pattern formation of 1D prey-taxis systems, J. Nonlinear Sci., 27 (2017), 71-97.  doi: 10.1007/s00332-016-9326-5.  Google Scholar

[29]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[30]

M. Winkler, Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations., 263 (2017), 4826-4869.  doi: 10.1016/j.jde.2017.06.002.  Google Scholar

[31]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differ. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[32]

S. WuJ. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations., 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024.  Google Scholar

[33] T. D. Wyatt, Pheromones and Animal Behaviour: Communication by Smell and Taste, Cambridge University Press, 2003.   Google Scholar
[34]

T. Xiang, Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics, Nonlinear Anal. Real World Appl., 39 (2018), 278-299.  doi: 10.1016/j.nonrwa.2017.07.001.  Google Scholar

[35]

M. Zuk and G. R. Kolluru, Exploitation of sexual signals by predators and parasitoids, Q. Rev. Biol., 73 (1998), 415-438.   Google Scholar

show all references

References:
[1]

I. Ahn and C. Yoon, Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis, J. Differential Equations, 268 (2020), 4222-4255.  doi: 10.1016/j.jde.2019.10.019.  Google Scholar

[2]

I. Ahn and C. Yoon, Global solvability of prey–predator models with indirect predator-taxis, Z. Angew. Math. Phys, 72 (2021), Paper No. 29, 20 pp. doi: 10.1007/s00033-020-01461-y.  Google Scholar

[3]

B. E. AinsebaM. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.  Google Scholar

[4]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.  Google Scholar

[5]

P. Amorim and B. Telch, A chemotaxis predator-prey model with indirect pursuit-evasion dynamics and parabolic signal, J. Math. Anal. Appl., 500 (2021), Paper No. 125128, 27 pp. doi: 10.1016/j.jmaa.2021.125128.  Google Scholar

[6]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

[7]

H. Bréezis and W. A. Strauss, Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[8]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity., 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[9]

A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, New York, 1969.  Google Scholar

[10]

X. He and S. Zheng, Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis, Appl. Math. Lett., 49 (2015), 73-77.  doi: 10.1016/j.aml.2015.04.017.  Google Scholar

[11]

C. D. HoeflerM. Taylor and E. M. Jakob, Chemosensory response to prey in pkidippus audax (araneae, salticidae) and pardosa milvina (araneae, lycosidae), J. Archnol., 30 (2002), 155-158.   Google Scholar

[12]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[13]

H.-Y. Jin and Z.-A. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.  Google Scholar

[14]

P. Kareiva and G. Odell, Swarms of predators exhibit preytaxis if individual predators use area-restricted search, Amer. Nat., 130 (1987), 233-270.  doi: 10.1086/284707.  Google Scholar

[15]

O. A. Ladyžzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, , Amer. Math. Soc. Transl., Vol. 23, Providence, RI, 1968.  Google Scholar

[16]

J. M. LeeT. Hillen and M. A. Lewis, Continuous traveling waves for prey-taxis, Bull. Math. Biol, 70 (2008), 654-676.  doi: 10.1007/s11538-007-9271-4.  Google Scholar

[17]

J. M. LeeT. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn, 3 (2009), 551-573.  doi: 10.1080/17513750802716112.  Google Scholar

[18]

G. LiY. Tao and M. Winkler, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst.-Ser. B., 25 (2020), 4383-4396.  doi: 10.3934/dcdsb.2020102.  Google Scholar

[19]

W. W. MurdochJ. Chesson and P. L. Chesson, Biological control in theory and practice, Amer. Nat., 125 (1985), 344-366.   Google Scholar

[20]

G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 545-582.  doi: 10.1016/j.nonrwa.2018.09.020.  Google Scholar

[21]

N. SapoukhinaY. Tyutyunov and R. Arditi, The role of prey taxis in biological control: A spatial theoretical model, Amer. Nat., 162 (2003), 61-76.  doi: 10.1086/375297.  Google Scholar

[22]

Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. Real World Appl., 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.  Google Scholar

[23]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[24]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[25]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst.-Ser. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar

[26]

J. Wang and M. Wang, The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741-6762.  doi: 10.1002/mma.5189.  Google Scholar

[27]

J. Wang and M. Wang, The dynamics of a predator-prey model with diffusion and indirect prey-taxis, J. Dyn. Diff. Equat., 32 (2020), 1291-1310.  doi: 10.1007/s10884-019-09778-7.  Google Scholar

[28]

Q. WangY. Song and L. Shao, Nonconstant positive steady states and pattern formation of 1D prey-taxis systems, J. Nonlinear Sci., 27 (2017), 71-97.  doi: 10.1007/s00332-016-9326-5.  Google Scholar

[29]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[30]

M. Winkler, Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations., 263 (2017), 4826-4869.  doi: 10.1016/j.jde.2017.06.002.  Google Scholar

[31]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differ. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[32]

S. WuJ. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations., 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024.  Google Scholar

[33] T. D. Wyatt, Pheromones and Animal Behaviour: Communication by Smell and Taste, Cambridge University Press, 2003.   Google Scholar
[34]

T. Xiang, Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics, Nonlinear Anal. Real World Appl., 39 (2018), 278-299.  doi: 10.1016/j.nonrwa.2017.07.001.  Google Scholar

[35]

M. Zuk and G. R. Kolluru, Exploitation of sexual signals by predators and parasitoids, Q. Rev. Biol., 73 (1998), 415-438.   Google Scholar

Figure 1.  Stabilization to the coexistence steady state
Figure 2.  Stabilization to the coexistence steady state
Figure 3.  Stabilization to the coexistence steady state
Figure 4.  Stabilization to the coexistence steady state
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