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doi: 10.3934/dcdsb.2021095

Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension

Qian Cao 1, , Yongli Cai 2,, and  Yong Luo 3,

1. 

School of Science, Chang'an University, Xi'an 710064, China
2. 

School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, China
3. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

* Corresponding author: Yongli Cai

Received  May 2020 Revised  December 2020 Early access  March 2021

Fund Project: The authors thank all referees for their careful reading and valuable feedback leading to an improvement of this paper. This research of Q. Cao was supported by the Fundamental Research Funds for the Central Universities (300102120103). This research of Y. Cai was supported by the National Science Foundation of China (No. 61672013 and 12071173)

Resorting to M.G. Crandall and P.H. Rabinowitz's well-known bifurcation theory we first obtain the local structure of steady states concerning the ratio–dependent predator–prey system with prey-taxis in spatial one dimension, which bifurcate from the homogeneous coexistence steady states when treating the prey–tactic coefficient as a bifurcation parameter. Based on this, then the global structure of positive solution is established. Moreover, through asymptotic analysis and eigenvalue perturbation we find the stability criterion of such bifurcating steady states. Finally, several numerical simulations are performed to show the pattern formation.

Keywords: Ratio–dependent, Predator–prey system, Prey–taxis, Global stability, Pattern formation.
Mathematics Subject Classification: Primary: 35A01, 35B40, 35K57, 92C15, 92C17.
Citation: Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021095
References:
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Y. Cai, Q. Cao and Z.-A. Wang, Asymptotic dynamics and spatial patterns of a ratio-dependent predator-prey system with prey-taxis, Applicable Analysis, 2020, 1–19. doi: 10.1080/00036811.2020.1728259.  Google Scholar

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A. Chakraborty, M. Singh, D. Lucy and P. Ridland, Predator-prey model with prey-taxis and diffusion, Mathematical and Computer Modelling, 46 (2007), 482-498.  doi: 10.1016/j.mcm.2006.10.010.  Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Archive for Rational Mechanics and Analysis, 52 (1973), 161-180.  doi: 10.1007/BF00282325.  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, Applied Mathematics Letters, 49 (2015), 73-77.  doi: 10.1016/j.aml.2015.04.017.  Google Scholar

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M. Hui Wang and Mark Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences, 171 (2001), 83-97.  doi: 10.1016/S0025-5564(01)00048-7.  Google Scholar

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J. Jang, W.-M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel–Epstein model, Journal of Dynamics and Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.  Google Scholar

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

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H.-Y. Jin and Z.-A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, European Journal of Applied Mathematics, (2020). Google Scholar

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J. M. Lee, T. Hillen and M. A. Lewis, Continuous traveling waves for prey-taxis, Bulletin of Mathematical Biology, 70 (2008), 654-676.  doi: 10.1007/s11538-007-9271-4.  Google Scholar

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J. M. Lee, T. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, Journal of Biological Dynamics, 3 (2009), 551-573.  doi: 10.1080/17513750802716112.  Google Scholar

[20]

C. Li, X. Wang and Y. Shao, Steady states of a predator-prey model with prey-taxis, Nonlinear Analysis: Theory, Methods & Applications, 97 (2014), 155-168.  doi: 10.1016/j.na.2013.11.022.  Google Scholar

[21]

M. Ma, C. Ou and Z.-A. Wang, Stationary solutions of a volume-filling chemotaxis model with logistic growth and their stability, SIAM Journal on Applied Mathematics, 72 (2012), 740-766.  doi: 10.1137/110843964.  Google Scholar

[22]

M. Ma and Z.-A. Wang, Global bifurcation and stability of steady states for a reaction-diffusion-chemotaxis model with volume-filling effect, Nonlinearity, 28 (2015), 2639-2660.  doi: 10.1088/0951-7715/28/8/2639.  Google Scholar

[23]

W. W. Murdoch, J. Chesson and P. L. Chesson, Biological control in theory and practice, The American Naturalist, 125 (1985), 344-366.  doi: 10.1086/284347.  Google Scholar

[24]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[25]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272.  Google Scholar

[26]

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

[27]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[28]

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

[29]

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

[30]

X. Wang, W. Wang and G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Mathematical Methods in the Applied Sciences, 38 (2015), 431-443.  doi: 10.1002/mma.3079.  Google Scholar

[31]

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

[32]

K. Yosida, Functional Analysis, 4th edition, Springer, Berlin, Heidelberg, 1974.  Google Scholar

show all references

References:
[1]

B. Ainseba, M. Bendahmane and A. Noussair, A reaction–diffusion system modeling predator–prey with prey-taxis, Nonlinear Analysis: Real World Applications, 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.  Google Scholar

[2]

H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent predation: An abstraction that works, Ecology, 76 (1995), 995-1004.  doi: 10.2307/1939362.  Google Scholar

[3]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, Journal of Theoretical Biology, 139 (1989), 311-326.   Google Scholar

[4]

R. Arditi, L. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent predation models, The American Naturalist, 138 (1991), 1287-1296.   Google Scholar

[5]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM Journal on Mathematical Analysis, 17 (1986), 1339-1353.  doi: 10.1137/0517094.  Google Scholar

[6]

Y. Cai, Q. Cao and Z.-A. Wang, Asymptotic dynamics and spatial patterns of a ratio-dependent predator-prey system with prey-taxis, Applicable Analysis, 2020, 1–19. doi: 10.1080/00036811.2020.1728259.  Google Scholar

[7]

A. Chakraborty, M. Singh, D. Lucy and P. Ridland, Predator-prey model with prey-taxis and diffusion, Mathematical and Computer Modelling, 46 (2007), 482-498.  doi: 10.1016/j.mcm.2006.10.010.  Google Scholar

[8]

C. Cosner, D. L. DeAngelis, J. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoretical Population Biology, 56 (1999), 65-75.   Google Scholar

[9]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[10]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Archive for Rational Mechanics and Analysis, 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[11]

D. Grünbaum, Using spatially explicit models to characterize foraging performance in heterogeneous landscapes, The American Naturalist, 151 (1998), 97-113.  doi: 10.1086/286105.  Google Scholar

[12]

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

[13]

M. Hui Wang and Mark Kot, Speeds of invasion in a model with strong or weak Allee effects, Mathematical Biosciences, 171 (2001), 83-97.  doi: 10.1016/S0025-5564(01)00048-7.  Google Scholar

[14]

J. Jang, W.-M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel–Epstein model, Journal of Dynamics and Differential Equations, 16 (2004), 297-320.  doi: 10.1007/s10884-004-2782-x.  Google Scholar

[15]

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

[16]

H.-Y. Jin and Z.-A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, European Journal of Applied Mathematics, (2020). Google Scholar

[17]

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

[18]

J. M. Lee, T. Hillen and M. A. Lewis, Continuous traveling waves for prey-taxis, Bulletin of Mathematical Biology, 70 (2008), 654-676.  doi: 10.1007/s11538-007-9271-4.  Google Scholar

[19]

J. M. Lee, T. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, Journal of Biological Dynamics, 3 (2009), 551-573.  doi: 10.1080/17513750802716112.  Google Scholar

[20]

C. Li, X. Wang and Y. Shao, Steady states of a predator-prey model with prey-taxis, Nonlinear Analysis: Theory, Methods & Applications, 97 (2014), 155-168.  doi: 10.1016/j.na.2013.11.022.  Google Scholar

[21]

M. Ma, C. Ou and Z.-A. Wang, Stationary solutions of a volume-filling chemotaxis model with logistic growth and their stability, SIAM Journal on Applied Mathematics, 72 (2012), 740-766.  doi: 10.1137/110843964.  Google Scholar

[22]

M. Ma and Z.-A. Wang, Global bifurcation and stability of steady states for a reaction-diffusion-chemotaxis model with volume-filling effect, Nonlinearity, 28 (2015), 2639-2660.  doi: 10.1088/0951-7715/28/8/2639.  Google Scholar

[23]

W. W. Murdoch, J. Chesson and P. L. Chesson, Biological control in theory and practice, The American Naturalist, 125 (1985), 344-366.  doi: 10.1086/284347.  Google Scholar

[24]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[25]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272.  Google Scholar

[26]

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

[27]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[28]

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

[29]

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

[30]

X. Wang, W. Wang and G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Mathematical Methods in the Applied Sciences, 38 (2015), 431-443.  doi: 10.1002/mma.3079.  Google Scholar

[31]

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

[32]

K. Yosida, Functional Analysis, 4th edition, Springer, Berlin, Heidelberg, 1974.  Google Scholar

Figure 1.  Pattern formation of system (6) where $ x\in(0,100) $ with grid number 1000 and $ t \in(0, 8000) $ with grid number 1000
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