January  2017, 37(1): 505-543. doi: 10.3934/dcds.2017021

Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

*Corresponding author

currently at Department of Mathematics, University of Central Florida, USA

Received  December 2015 Revised  August 2016 Published  November 2016

Fund Project: QW is supported by NSF-China (Grant No. 11501460) and the Project (No.15ZA0382) from Department of Education, Sichuan, China.

This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension $ N=1,2 $ and for the parabolic-parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that prey-taxis stabilizes the positive constant equilibrium if there is no group defense while it destabilizes the equilibrium otherwise. Then we obtain stationary and time-periodic nontrivial solutions of the system that bifurcate from the positive constant equilibrium. Moreover, the stability of these solutions is also analyzed in detail which provides a wave mode selection mechanism of nontrivial patterns for this strongly coupled system. Finally, we perform numerical simulations to illustrate and support our theoretical results.

Citation: Ke Wang, Qi Wang, Feng Yu. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021
References:
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P. Abrams and H. Matsuda, Effects of adaptive predatory and anti-predator behaviour in a two-prey-one-predator system, Evolutionary Ecology, 7 (1993), 312-326.  doi: 10.1007/BF01237749.

[2]

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.

[3]

N. Alikakos, $ L^p $ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[4]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. 

[5]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, Delay Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, 1475 (1991), 53-63.  doi: 10.1007/BFb0083479.

[6]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.

[7]

A. ChakrabortyM. SinghD. Lucy and P. Ridland, Predator-prey model with prey-taxis and diffusion, Math. Comput. Modelling, 46 (2007), 482-498.  doi: 10.1016/j.mcm.2006.10.010.

[8]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.

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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.

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M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72.  doi: 10.1007/BF00280827.

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W. D. Hamilton, Geometry for the selfish herd, J. Theoret. Biol., 31 (1971), 295-311.  doi: 10.1016/0022-5193(71)90189-5.

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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, Appl. Math. Lett., 49 (2015), 73-77.  doi: 10.1016/j.aml.2015.04.017.

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L. Hsiao and P. de Mottoni, Persistence in reacting-diffusing systems: Interaction of two predators and one prey, Nonlinear Anal., 11 (1987), 877-891.  doi: 10.1016/0362-546X(87)90058-7.

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L. Jin, Q. Wang and Z. Zhang, Qualitative Studies of Advective Competition System with Beddington-DeAngelis Functional Response, preprint, http://arxiv.org/abs/1412.3371

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T. Kato, Functional Analysis, Springer Classics in Mathematics, 1995. doi: 10.1007/978-3-642-61859-8.

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

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K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 293-314.  doi: 10.1016/j.jde.2003.10.016.

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K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.  doi: 10.1016/j.jde.2003.08.003.

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O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968,648 pages.

[32]

J. M. LeeT. Hilllen and M. A. Lewis, Continuous traveling waves for prey-taxis, Bull. Math. Biol., 70 (2008), 654-676.  doi: 10.1007/s11538-007-9271-4.

[33]

J. M. LeeT. Hilllen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.  doi: 10.1080/17513750802716112.

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C. LiX. Wang and Y. Shao, Steady states of a predator-prey model with prey-taxis, Nonlinear Anal., 97 (2014), 155-168.  doi: 10.1016/j.na.2013.11.022.

[35]

J.-J. LinW. WangC. Zhao and T.-H. Yang, Global dynamics and traveling wave solutions of two predators-one prey models, Discrete Contin. Dyn. Syst-Series B, 20 (2015), 1135-1154.  doi: 10.3934/dcdsb.2015.20.1135.

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

[37]

I. LoladzeY. KuangJ.-J. Elser and W.-F. Fagan, Competition and stoichiometry: Coexistence of two predators on one prey, Theoret. Pop. Biol., 65 (2004), 1-15.  doi: 10.1016/S0040-5809(03)00105-9.

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Z. Maciej GliwiczP. MaszczykJ. Jabłoński and D. Wrzosek, Patch exploitation by planktivorous fish and the concept of aggregation as an antipredation defense in zooplankton, Limnology and Oceanography, 58 (2013), 1621-1639.  doi: 10.4319/lo.2013.58.5.1621.

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W. Nagata and S.-M. Merchant, Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.  doi: 10.1016/j.physd.2010.04.014.

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P. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.

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C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.

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

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K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst., 9 (2003), 1049-1061.  doi: 10.3934/dcds.2003.9.1049.

[49]

N. SapoukhinaY. 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.

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J. Shi and X. 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.

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N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

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G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. 

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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.

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show all references

References:
[1]

P. Abrams and H. Matsuda, Effects of adaptive predatory and anti-predator behaviour in a two-prey-one-predator system, Evolutionary Ecology, 7 (1993), 312-326.  doi: 10.1007/BF01237749.

[2]

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.

[3]

N. Alikakos, $ L^p $ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[4]

H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. 

[5]

H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, Delay Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, 1475 (1991), 53-63.  doi: 10.1007/BFb0083479.

[6]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, differential operators and nonlinear Analysis, Teubner, Stuttgart, Leipzig, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.

[7]

A. ChakrabortyM. SinghD. Lucy and P. Ridland, Predator-prey model with prey-taxis and diffusion, Math. Comput. Modelling, 46 (2007), 482-498.  doi: 10.1016/j.mcm.2006.10.010.

[8]

A. ChertockA. KurganovX. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.  doi: 10.3934/krm.2012.5.51.

[9]

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.

[10]

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.

[11]

M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72.  doi: 10.1007/BF00280827.

[12]

T. Czaran, Spatiotemporal Models of Population and Community Dynamics, Chapman and Hall, London, 1998.

[13]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[14]

D. Grünbaum, Advection-diffusion equations for generalized tactic searching behaviours, J. Math. Biol., 38 (1999), 169-194.  doi: 10.1007/s002850050145.

[15]

W. D. Hamilton, Geometry for the selfish herd, J. Theoret. Biol., 31 (1971), 295-311.  doi: 10.1016/0022-5193(71)90189-5.

[16]

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.

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981. doi: 10.1007/BFb0089647.

[18]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[19]

D. Horstmann, 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber DMV, 105 (2003), 103-165. 

[20]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.

[21]

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.

[22]

L. Hsiao and P. de Mottoni, Persistence in reacting-diffusing systems: Interaction of two predators and one prey, Nonlinear Anal., 11 (1987), 877-891.  doi: 10.1016/0362-546X(87)90058-7.

[23]

L. Jin, Q. Wang and Z. Zhang, Qualitative Studies of Advective Competition System with Beddington-DeAngelis Functional Response, preprint, http://arxiv.org/abs/1412.3371

[24]

D. D. Joseph and D. Nield, Stability of bifurcating time-periodic and steady solutions of arbitrary amplitude, Arch. Rational Mech. Anal., 58 (1975), 369-380.  doi: 10.1007/BF00250296.

[25]

D. D. Joseph and D. H. Sattinger, Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal., 45 (1972), 75-109.  doi: 10.1007/BF00253039.

[26]

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.

[27]

T. Kato, Functional Analysis, Springer Classics in Mathematics, 1995. doi: 10.1007/978-3-642-61859-8.

[28]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[29]

K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 293-314.  doi: 10.1016/j.jde.2003.10.016.

[30]

K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348.  doi: 10.1016/j.jde.2003.08.003.

[31]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968,648 pages.

[32]

J. M. LeeT. Hilllen and M. A. Lewis, Continuous traveling waves for prey-taxis, Bull. Math. Biol., 70 (2008), 654-676.  doi: 10.1007/s11538-007-9271-4.

[33]

J. M. LeeT. Hilllen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573.  doi: 10.1080/17513750802716112.

[34]

C. LiX. Wang and Y. Shao, Steady states of a predator-prey model with prey-taxis, Nonlinear Anal., 97 (2014), 155-168.  doi: 10.1016/j.na.2013.11.022.

[35]

J.-J. LinW. WangC. Zhao and T.-H. Yang, Global dynamics and traveling wave solutions of two predators-one prey models, Discrete Contin. Dyn. Syst-Series B, 20 (2015), 1135-1154.  doi: 10.3934/dcdsb.2015.20.1135.

[36]

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

[37]

I. LoladzeY. KuangJ.-J. Elser and W.-F. Fagan, Competition and stoichiometry: Coexistence of two predators on one prey, Theoret. Pop. Biol., 65 (2004), 1-15.  doi: 10.1016/S0040-5809(03)00105-9.

[38]

Z. Maciej GliwiczP. MaszczykJ. Jabłoński and D. Wrzosek, Patch exploitation by planktivorous fish and the concept of aggregation as an antipredation defense in zooplankton, Limnology and Oceanography, 58 (2013), 1621-1639.  doi: 10.4319/lo.2013.58.5.1621.

[39]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64.  doi: 10.1007/BF00276035.

[40]

J. D. Murray, Mathematical Biology, Springer, New York, 1993. doi: 10.1007/b98869.

[41]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.  doi: 10.1155/AAA/2006/23061.

[42]

W. Nagata and S.-M. Merchant, Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.  doi: 10.1016/j.physd.2010.04.014.

[43]

K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122. 

[44]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems, Modern Perspectives 2nd Edition, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.

[45]

P. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.

[46]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.

[47]

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

[48]

K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst., 9 (2003), 1049-1061.  doi: 10.3934/dcds.2003.9.1049.

[49]

N. SapoukhinaY. 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.

[50]

J. Shi and X. 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.

[51]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[52]

G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. 

[53]

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.

[54]

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.

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Figure 1.  Pitch-fork bifurcation diagrams when case (ⅰ) in Theorem 4.3 occurs. The stable bifurcation curve is plotted in solid line and the unstable bifurcation curve is plotted in dashed line. The branch $\Gamma_{k_0}(s)$ around $(\bar u, \bar v, \bar w, \chi^S_{k_0})$ is stable if it turns to the right and is unstable if it turns to the left, while $\Gamma_{k}(s)$ around $(\bar u, \bar v, \bar w, \chi^S_{k})$ is always unstable if $k\neq k_0$
Figure 2.  Formation of stationary patterns of (3.1) over $\Omega=(0, 7)$. System parameters in all the graphes are taken to be the same as in Table 1 except that $\chi=8$, which is larger than $\chi^S_{k_0}\approx6.05$ given in Table 1. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$, while the stable pattern has wave mode $\cos \frac{6\pi x}{7}$. These graphes support our stability analysis of the bifurcating solutions
Figure 3.  Formation of stationary patterns of (3.1) over intervals with lengthes $L=9$, $11$, $13$ and $15$. System parameters here are taken to be the same as those in Table 1 except that $\chi=8$, which is slightly larger than $\chi^S_{k_0}\approx6.04$ given in Table 2. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$. These graphes support our stability analysis of the bifurcating solutions and indicate that large intervals support more aggregates than small intervals
Figure 4.  Formation of time-periodic spatial patterns of (3.1) over $\Omega=(0, 7)$. System parameters in all the graphes are taken to be the same as in Table 3 except that $\chi=120$, which is slightly larger than $\chi^H_{k_0}\approx 92.57$ given in Table 3. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$, however the stable oscillating patterns have spatial profile $\cos\frac{3\pi x}{7}$, which emerge periodically. These plots support our stability analysis in Section 5
Figure 5.  Formation of time-periodic spatial patterns of (3.1) over intervals with lengthes $L=9$, $11$, $13$ and $15$ respectively. System parameters in all the graphes are taken to be the same as in Table 3 except that $\chi=120$, which is slightly larger than $\chi^H_{k_0}$ given in Table 4. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$. These graphes support our stability analysis of the bifurcating solutions
Figure 6.  Formation and development of boundary spike through wave propagation. Initial data are $(u_0, v_0, w_0)=(\bar u, \bar v, \bar w)+(0.01, 0.01, 0.01)\cos \pi x$. Prey-taxis rate $\chi=3000$ which is far away from the critical bifurcation value $\chi_0=956.79$.
Figure 7.  Pattern formations in (3.1) due to the effect of large prey-taxis rate $\chi$. Various interesting and complex spatial-temporal dynamics are observed in this system
Table 1.  Values of $\chi^S_k$ in (3.4) and $\chi^H_k$ in (3.5) for $L=7$. The system parameters are chosen to be $d_1=d_3=0.1$, $d_2=2$, $\alpha_1=\beta_1=\beta_2=0.5$, $\alpha_2=2$, $\alpha_3=1$ and $\beta_{31}=\beta_{32}=0.1$, $\xi=0.5$. The sensitivity function is $\phi(w)=w(0.1-w)$ which models the group defense of the preys when population density surpasses 0.1. We see that $\min_{k\in\mathbb N^+}\{\chi^S_k, \chi^H_k\}=\chi^S_6$. Therefore $(\bar u, \bar v, \bar w)$ loses its stability to the stable wave mode $\cos \frac{6\pi x}{7}$. This is numerically verified in Figure 2
$k$12345 6789
$\chi^S_{k}$66.9818.9810.307.496.416.056.096.366.80
$\chi^H_{k}$1204.20550.84504.20575.80705.50878.481089.701336.901619.19
$k$12345 6789
$\chi^S_{k}$66.9818.9810.307.496.416.056.096.366.80
$\chi^H_{k}$1204.20550.84504.20575.80705.50878.481089.701336.901619.19
Table 2.  Stable wave mode numbers and the corresponding bifurcation values $\chi_0$ for different interval lengthes. System parameters are chosen to be the same as those in Table 1. We see that the threshold value $\chi_0$ is always achieved at the steady state bifurcation point $\chi^S_{k_0}$. This table also indicates that larger intervals support higher wave modes
Interval length $L$12345678
$k_0$12345567
$\chi_0=\chi^S_{k_0}$6.096.096.096.096.096.086.056.04
Interval length $L$910111213141516
$k_0$89101112131415
$\chi_0=\chi^S_{k_0}$6.046.036.046.046.046.046.046.04
Interval length $L$12345678
$k_0$12345567
$\chi_0=\chi^S_{k_0}$6.096.096.096.096.096.086.056.04
Interval length $L$910111213141516
$k_0$89101112131415
$\chi_0=\chi^S_{k_0}$6.046.036.046.046.046.046.046.04
Table 3.  Values of $\chi^S_k$ in (3.4) and $\chi^H_k$ in (3.5) for $L=7$. System parameters are $d_1=d_3=1$, $d_2=0.01$, $\alpha_1=0.02$, $\alpha_2=0.04$, $\alpha_3=8$ and $\beta_1=0.05$, $\beta_2=\beta_{31}=\beta_{32}=0.5$, while the sensitivity function is $\phi(w)=w(0.2-w)$. We see that $\min_{k\in\mathbb N^+}\{\chi^S_k, \chi^H_k\}=\chi^H_3$. Therefore $(\bar u, \bar v, \bar w)$ loses its stability to the time-periodic solutions with wave mode $\cos \frac{3\pi x}{7}$. This is numerically verified in Figure 4
$k$12 3456789
$\chi^S_{k}$106.498.63107.32122.53143.15169.05200.24236.80278.76
$\chi^H_{k}$186.3796.7392.57105.46127.03155.24189.40229.24274.64
$k$12 3456789
$\chi^S_{k}$106.498.63107.32122.53143.15169.05200.24236.80278.76
$\chi^H_{k}$186.3796.7392.57105.46127.03155.24189.40229.24274.64
Table 4.  Stable wave mode numbers and the corresponding bifurcation values $\chi_0$ for different interval lengthes, where system parameters are chosen to be the same as those in Table 3. We see that the threshold value $\chi_0$ is always achieved at the Hopf bifurcation point $\chi^H_{k_0}$
Interval length $L$23456789
$k_0$12345567
$\chi_0=\chi^H_{k_0}$97.6892.1397.6891.4992.1392.5791.1592.13
Interval length $L$1011121314151617
$k_0$89101112131415
$\chi_0=\chi^H_{k_0}$91.591.291.1391.2191.3091.4991.1591.40
Interval length $L$23456789
$k_0$12345567
$\chi_0=\chi^H_{k_0}$97.6892.1397.6891.4992.1392.5791.1592.13
Interval length $L$1011121314151617
$k_0$89101112131415
$\chi_0=\chi^H_{k_0}$91.591.291.1391.2191.3091.4991.1591.40
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