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March  2021, 26(3): 1273-1289. doi: 10.3934/dcdsb.2020162

Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis

1. 

School of Mathematics Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China

2. 

College of Science, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu, 210003, China

3. 

Department of Mathematics, William & Mary, Williamsburg, Virginia 23187-8795, USA

* Corresponding author: Junping Shi

Received  February 2019 Revised  March 2020 Published  May 2020

Fund Project: Partially supported by a grant from China Scholarship Council, US-NSF grant DMS-1715651, National Natural Science Foundation of China grant 11971135 and National Natural Science Foundation of Heilongjiang Province grant LH2019A017, 2018-KYYWF-0999

A reaction-diffusion predator-prey system with prey-taxis and predator-taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing prey and prey evading predators. The spatial pattern formation induced by the prey-taxis and predator-taxis is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. It is shown that both attractive prey-taxis and repulsive predator-taxis compress the spatial patterns, while repulsive prey-taxis and attractive predator-taxis help to generate spatial patterns. Our results are applied to the Holling-Tanner predator-prey model to demonstrate the pattern formation mechanism.

Citation: Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162
References:
[1]

B. AinsebaM. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.   Google Scholar

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems. Ⅲ. Global existence, Math. Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar

[3]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differ. Integral Equ., 3 (1990), 13-75.   Google Scholar

[4]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.  Google Scholar

[5]

S. S. Chen and J. P. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 25 (2012), 614-618.  doi: 10.1016/j.aml.2011.09.070.  Google Scholar

[6]

S. S. Chen, J. P. Shi and J. J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250061, 11 pp. doi: 10.1142/S0218127412500617.  Google Scholar

[7]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.  doi: 10.3934/dcds.2014.34.1701.  Google Scholar

[8]

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

[9]

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

[10]

P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z., 194 (1987), 375-396.  doi: 10.1007/BF01162244.  Google Scholar

[11]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[12]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[13]

S.-B. Hsu and T.-W. Hwang, Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.   Google Scholar

[14]

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

[15]

J. Y. JinJ. P. ShiJ. J. Wei and F. Q. Yi, Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637.  Google Scholar

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P. Kareiva and G. T. Odell, Swarms of predators exhibit " preytaxis" if individual predators use area-restricted search, Am. Nat., 130 (1987), 233-270.   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]

X. LiW. H. Jiang and J. P. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.  Google Scholar

[19]

P. LiuJ. P. Shi and Z.-A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.  Google Scholar

[20]

R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902.  doi: 10.1126/science.177.4052.900.  Google Scholar

[21]

J. D. Murray, Mathematical Biology. I: An Introduction, Third edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[22]

R. Peng and M. X. Wang, Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Appl. Math. Lett., 20 (2007), 664-670.  doi: 10.1016/j.aml.2006.08.020.  Google Scholar

[23]

Y. W. Qi and Y. Zhu, The study of global stability of a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 57 (2016), 132-138.  doi: 10.1016/j.aml.2016.01.017.  Google Scholar

[24]

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

[25]

J. P. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[26]

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

[27]

J. P. ShiZ. F. Xie and K. Little, Cross-diffusion induced instability and stability in reaction-diffusion systems, J. Appl. Anal. Comput., 1 (2011), 95-119.   Google Scholar

[28]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.  Google Scholar

[29]

Y. S. 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

[30]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[31]

A. M. Turner and G. G. Mittelbach, Predator avoidance and community structure: Interactions among piscivores, planktivores, and plankton, Ecology, 71 (1990), 2241-2254.  doi: 10.2307/1938636.  Google Scholar

[32]

J. F. WangJ. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[33]

J. F. WangJ. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.   Google Scholar

[34]

J. F. Wang and Y. W. Wang, Bifurcation analysis in a diffusive Segel-Jackson model, J. Math. Anal. Appl., 415 (2014), 204-216.  doi: 10.1016/j.jmaa.2014.01.070.  Google Scholar

[35]

J. F. WangJ. J. Wei and J. P. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.  Google Scholar

[36]

Q. WangY. Song and L. J. 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

[37]

W. M. Wang, Z. G. Guo, R. K. Upadhyay and Y. Z. Lin, Pattern formation in a cross-diffusive Holling-Tanner model, Discrete Dyn. Nat. Soc., (2012), Art. ID 828219, 12 pp. doi: 10.1155/2012/828219.  Google Scholar

[38]

X. L. WangW. D. Wang and G. H. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Math. Methods Appl. Sci., 38 (2015), 431-443.  doi: 10.1002/mma.3079.  Google Scholar

[39]

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

[40]

M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^1$, Adv. Nonlinear Anal., 9 (2020), 526-566.  doi: 10.1515/anona-2020-0013.  Google Scholar

[41]

S. N. WuJ. P. Shi and B. Y. 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

[42]

S. N. WuJ. F. Wang and J. P. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.  doi: 10.1142/S0218202518400158.  Google Scholar

[43]

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

[44]

F. Q. YiJ. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

[45]

T. M. Zaret and J. S. Suffern, Vertical migration in zooplankton as a predator avoidance mechanism, Limnology and Oceanography, 21 (1976), 804-813.   Google Scholar

show all references

References:
[1]

B. AinsebaM. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9 (2008), 2086-2105.   Google Scholar

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems. Ⅲ. Global existence, Math. Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar

[3]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differ. Integral Equ., 3 (1990), 13-75.   Google Scholar

[4]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.  Google Scholar

[5]

S. S. Chen and J. P. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 25 (2012), 614-618.  doi: 10.1016/j.aml.2011.09.070.  Google Scholar

[6]

S. S. Chen, J. P. Shi and J. J. Wei, Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250061, 11 pp. doi: 10.1142/S0218127412500617.  Google Scholar

[7]

C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.  doi: 10.3934/dcds.2014.34.1701.  Google Scholar

[8]

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

[9]

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

[10]

P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z., 194 (1987), 375-396.  doi: 10.1007/BF01162244.  Google Scholar

[11]

Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[12]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.  Google Scholar

[13]

S.-B. Hsu and T.-W. Hwang, Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.   Google Scholar

[14]

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

[15]

J. Y. JinJ. P. ShiJ. J. Wei and F. Q. Yi, Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637.  Google Scholar

[16]

P. Kareiva and G. T. Odell, Swarms of predators exhibit " preytaxis" if individual predators use area-restricted search, Am. Nat., 130 (1987), 233-270.   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]

X. LiW. H. Jiang and J. P. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287-306.  doi: 10.1093/imamat/hxr050.  Google Scholar

[19]

P. LiuJ. P. Shi and Z.-A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.  Google Scholar

[20]

R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902.  doi: 10.1126/science.177.4052.900.  Google Scholar

[21]

J. D. Murray, Mathematical Biology. I: An Introduction, Third edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[22]

R. Peng and M. X. Wang, Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model, Appl. Math. Lett., 20 (2007), 664-670.  doi: 10.1016/j.aml.2006.08.020.  Google Scholar

[23]

Y. W. Qi and Y. Zhu, The study of global stability of a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 57 (2016), 132-138.  doi: 10.1016/j.aml.2016.01.017.  Google Scholar

[24]

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

[25]

J. P. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[26]

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

[27]

J. P. ShiZ. F. Xie and K. Little, Cross-diffusion induced instability and stability in reaction-diffusion systems, J. Appl. Anal. Comput., 1 (2011), 95-119.   Google Scholar

[28]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.  Google Scholar

[29]

Y. S. 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

[30]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[31]

A. M. Turner and G. G. Mittelbach, Predator avoidance and community structure: Interactions among piscivores, planktivores, and plankton, Ecology, 71 (1990), 2241-2254.  doi: 10.2307/1938636.  Google Scholar

[32]

J. F. WangJ. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[33]

J. F. WangJ. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.   Google Scholar

[34]

J. F. Wang and Y. W. Wang, Bifurcation analysis in a diffusive Segel-Jackson model, J. Math. Anal. Appl., 415 (2014), 204-216.  doi: 10.1016/j.jmaa.2014.01.070.  Google Scholar

[35]

J. F. WangJ. J. Wei and J. P. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.  Google Scholar

[36]

Q. WangY. Song and L. J. 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

[37]

W. M. Wang, Z. G. Guo, R. K. Upadhyay and Y. Z. Lin, Pattern formation in a cross-diffusive Holling-Tanner model, Discrete Dyn. Nat. Soc., (2012), Art. ID 828219, 12 pp. doi: 10.1155/2012/828219.  Google Scholar

[38]

X. L. WangW. D. Wang and G. H. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Math. Methods Appl. Sci., 38 (2015), 431-443.  doi: 10.1002/mma.3079.  Google Scholar

[39]

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

[40]

M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^1$, Adv. Nonlinear Anal., 9 (2020), 526-566.  doi: 10.1515/anona-2020-0013.  Google Scholar

[41]

S. N. WuJ. P. Shi and B. Y. 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

[42]

S. N. WuJ. F. Wang and J. P. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.  doi: 10.1142/S0218202518400158.  Google Scholar

[43]

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

[44]

F. Q. YiJ. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

[45]

T. M. Zaret and J. S. Suffern, Vertical migration in zooplankton as a predator avoidance mechanism, Limnology and Oceanography, 21 (1976), 804-813.   Google Scholar

Figure 1.  The stable region/unstable region of equilibrium $ (u^*, v^*) $ of system (1) in $ \xi-\eta $ parameter plane. Here $ f, g $ and other parameters are taken from (25) in Section 4. (Left): $ d = 0.06 $, $ (0, 0)\in S $; (Right): $ d = 0.01 $, $ (0, 0)\not\in S $. The thick solid curve is $ f_u+dg_v+\eta v^*f_v-\xi u^*g_u-2\sqrt{d+\xi\eta u^*v^*}\sqrt{Det J} = 0 $, the thin solid curve is $ d+\xi\eta u^*v^* = 0 $ in both panels
Figure 2.  (Left): $ D(\eta, \xi, p) = 0 $ in $ \xi-p $ plane for fixed $ \eta $; (right): there are two bifurcation value $ \xi_S^2 $(corresponding to $ \lambda_2 = 4 $) and $ \xi_S^3 $(corresponding to $ \lambda_3 = 9 $), here $ f, g $ and other parameters are taken from (25) in Section 4 with $ \eta = 0.02 $, $ d = 0.01 $, $ \Omega = (0, \pi) $
Figure 3.  Turing instability for (25) with $ \beta = 0.2 $, $ m = 2 $, $ s = 0.5 $, $ \Omega = (0, 10\pi) $, $ d = 0.01 $, $ \xi = \eta = 0 $ and initial value $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $
Figure 4.  Stable constant equilibrium for (25) with $ \beta = 0.2 $, $ m = 2 $, $ s = 0.5 $, $ \Omega = (0, 10\pi) $, $ d = 0.06 $, $ \xi = \eta = 0 $ and initial value $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $
Figure 5.  Turing pattern in (25) is stabilized when $ \xi = 0.9 $ and $ \eta = 0.4 $, and the same initial value $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $. Other parameters are also same as in Figure 3
Figure 6.  Instability induced by taxis in (25) when $ \xi = -0.5 $ and $ \eta = -0.4 $ and initial condition $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $. Other parameters are same as in Figure 4
Figure 7.  The amplitude change of non-constant equilibrium solutions of (25) for different $ (\xi, \eta) $. Here the value is the difference of maximum and minimum values of the non-constant equilibrium solution; the initial value is $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $ and other parameters are the same as Figure 3
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