September  2022, 27(9): 4855-4874. doi: 10.3934/dcdsb.2021255

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 Published  September 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (9) : 4855-4874. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[35]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[35]

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

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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