# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021255
##### References:
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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. Hoefler, M. 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. Lee, T. 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. Lee, T. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn, 3 (2009), 551-573.  doi: 10.1080/17513750802716112. [18] G. Li, Y. 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. Murdoch, J. 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. Sapoukhina, Y. 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. Wang, Y. 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. Wu, J. 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. Ainseba, M. 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. Hoefler, M. 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. Lee, T. 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. Lee, T. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn, 3 (2009), 551-573.  doi: 10.1080/17513750802716112. [18] G. Li, Y. 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. Murdoch, J. 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. Sapoukhina, Y. 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. Wang, Y. 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. Wu, J. 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.
Stabilization to the coexistence steady state
Stabilization to the coexistence steady state
Stabilization to the coexistence steady state
Stabilization to the coexistence steady state
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