# American Institute of Mathematical Sciences

March  2022, 27(3): 1397-1420. doi: 10.3934/dcdsb.2021095

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

 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 Published  March 2022 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.

Citation: Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1397-1420. doi: 10.3934/dcdsb.2021095
##### 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. [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. [3] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [32] K. Yosida, Functional Analysis, 4th edition, Springer, Berlin, Heidelberg, 1974.

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##### 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. [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. [3] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [32] K. Yosida, Functional Analysis, 4th edition, Springer, Berlin, Heidelberg, 1974.
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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