# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021
##### 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. Google Scholar [2] 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. Google Scholar [3] N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. Google Scholar [4] H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. Google Scholar [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. Google Scholar [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. Google Scholar [7] A. Chakraborty, M. Singh, D. 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. Google Scholar [8] A. Chertock, A. Kurganov, X. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] T. Czaran, Spatiotemporal Models of Population and Community Dynamics, Chapman and Hall, London, 1998.Google Scholar [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. Google Scholar [14] D. Grünbaum, Advection-diffusion equations for generalized tactic searching behaviours, J. Math. Biol., 38 (1999), 169-194. doi: 10.1007/s002850050145. Google Scholar [15] W. D. Hamilton, Geometry for the selfish herd, J. Theoret. Biol., 31 (1971), 295-311. doi: 10.1016/0022-5193(71)90189-5. Google Scholar [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. Google Scholar [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981. doi: 10.1007/BFb0089647. Google Scholar [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. Google Scholar [19] D. Horstmann, 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber DMV, 105 (2003), 103-165. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] L. Jin, Q. Wang and Z. Zhang, Qualitative Studies of Advective Competition System with Beddington-DeAngelis Functional Response, preprint, http://arxiv.org/abs/1412.3371Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] T. Kato, Functional Analysis, Springer Classics in Mathematics, 1995. doi: 10.1007/978-3-642-61859-8. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] J. M. Lee, T. 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. Google Scholar [33] J. M. Lee, T. Hilllen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573. doi: 10.1080/17513750802716112. Google Scholar [34] C. Li, X. 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. Google Scholar [35] J.-J. Lin, W. Wang, C. 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. Google Scholar [36] P. Liu, J. 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. Google Scholar [37] I. Loladze, Y. Kuang, J.-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. Google Scholar [38] Z. Maciej Gliwicz, P. Maszczyk, J. 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. Google Scholar [39] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035. Google Scholar [40] J. D. Murray, Mathematical Biology, Springer, New York, 1993. doi: 10.1007/b98869. Google Scholar [41] T. Nagai, T. 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. Google Scholar [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. Google Scholar [43] K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122. Google Scholar [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. Google Scholar [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. Google Scholar [46] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar [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. Google Scholar [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. Google Scholar [49] 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 [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. Google Scholar [51] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. Google Scholar [52] G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. Google Scholar [53] Y. 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##### 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. Google Scholar [2] 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. Google Scholar [3] N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. Google Scholar [4] H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. Google Scholar [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. Google Scholar [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. Google Scholar [7] A. Chakraborty, M. Singh, D. 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. Google Scholar [8] A. Chertock, A. Kurganov, X. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] T. Czaran, Spatiotemporal Models of Population and Community Dynamics, Chapman and Hall, London, 1998.Google Scholar [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. Google Scholar [14] D. Grünbaum, Advection-diffusion equations for generalized tactic searching behaviours, J. Math. Biol., 38 (1999), 169-194. doi: 10.1007/s002850050145. Google Scholar [15] W. D. Hamilton, Geometry for the selfish herd, J. Theoret. Biol., 31 (1971), 295-311. doi: 10.1016/0022-5193(71)90189-5. Google Scholar [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. Google Scholar [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981. doi: 10.1007/BFb0089647. Google Scholar [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. Google Scholar [19] D. Horstmann, 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber DMV, 105 (2003), 103-165. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] L. Jin, Q. Wang and Z. Zhang, Qualitative Studies of Advective Competition System with Beddington-DeAngelis Functional Response, preprint, http://arxiv.org/abs/1412.3371Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] T. Kato, Functional Analysis, Springer Classics in Mathematics, 1995. doi: 10.1007/978-3-642-61859-8. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] J. M. Lee, T. 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. Google Scholar [33] J. M. Lee, T. Hilllen and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3 (2009), 551-573. doi: 10.1080/17513750802716112. Google Scholar [34] C. Li, X. 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. Google Scholar [35] J.-J. Lin, W. Wang, C. 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. Google Scholar [36] P. Liu, J. 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. Google Scholar [37] I. Loladze, Y. Kuang, J.-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. Google Scholar [38] Z. Maciej Gliwicz, P. Maszczyk, J. 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. Google Scholar [39] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035. Google Scholar [40] J. D. Murray, Mathematical Biology, Springer, New York, 1993. doi: 10.1007/b98869. Google Scholar [41] T. Nagai, T. 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. Google Scholar [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. Google Scholar [43] K. Nakashima and Y. Yamada, Positive steady states for prey-predator models with cross-diffusion, Adv. Differential Equations, 1 (1996), 1099-1122. Google Scholar [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. Google Scholar [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. Google Scholar [46] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar [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. Google Scholar [48] K. Ryu and I. 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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 $k\neq k_0$
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
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
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
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
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 $\chi_0=956.79$.
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
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
 $k$ 1 2 3 4 5 6 7 8 9 $\chi^S_{k}$ 66.98 18.98 10.3 7.49 6.41 6.05 6.09 6.36 6.8 $\chi^H_{k}$ 1204.2 550.84 504.2 575.8 705.5 878.48 1089.7 1336.9 1619.19
 $k$ 1 2 3 4 5 6 7 8 9 $\chi^S_{k}$ 66.98 18.98 10.3 7.49 6.41 6.05 6.09 6.36 6.8 $\chi^H_{k}$ 1204.2 550.84 504.2 575.8 705.5 878.48 1089.7 1336.9 1619.19
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 $L$ 1 2 3 4 5 6 7 8 $k_0$ 1 2 3 4 5 5 6 7 $\chi_0=\chi^S_{k_0}$ 6.09 6.09 6.09 6.09 6.09 6.08 6.05 6.04 Interval length $L$ 9 10 11 12 13 14 15 16 $k_0$ 8 9 10 11 12 13 14 15 $\chi_0=\chi^S_{k_0}$ 6.04 6.03 6.04 6.04 6.04 6.04 6.04 6.04
 Interval length $L$ 1 2 3 4 5 6 7 8 $k_0$ 1 2 3 4 5 5 6 7 $\chi_0=\chi^S_{k_0}$ 6.09 6.09 6.09 6.09 6.09 6.08 6.05 6.04 Interval length $L$ 9 10 11 12 13 14 15 16 $k_0$ 8 9 10 11 12 13 14 15 $\chi_0=\chi^S_{k_0}$ 6.04 6.03 6.04 6.04 6.04 6.04 6.04 6.04
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
 $k$ 1 2 3 4 5 6 7 8 9 $\chi^S_{k}$ 106.4 98.63 107.32 122.53 143.15 169.05 200.24 236.8 278.76 $\chi^H_{k}$ 186.37 96.73 92.57 105.46 127.03 155.24 189.4 229.24 274.64
 $k$ 1 2 3 4 5 6 7 8 9 $\chi^S_{k}$ 106.4 98.63 107.32 122.53 143.15 169.05 200.24 236.8 278.76 $\chi^H_{k}$ 186.37 96.73 92.57 105.46 127.03 155.24 189.4 229.24 274.64
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 $L$ 2 3 4 5 6 7 8 9 $k_0$ 1 2 3 4 5 5 6 7 $\chi_0=\chi^H_{k_0}$ 97.68 92.13 97.68 91.49 92.13 92.57 91.15 92.13 Interval length $L$ 10 11 12 13 14 15 16 17 $k_0$ 8 9 10 11 12 13 14 15 $\chi_0=\chi^H_{k_0}$ 91.5 91.2 91.13 91.21 91.3 91.49 91.15 91.4
 Interval length $L$ 2 3 4 5 6 7 8 9 $k_0$ 1 2 3 4 5 5 6 7 $\chi_0=\chi^H_{k_0}$ 97.68 92.13 97.68 91.49 92.13 92.57 91.15 92.13 Interval length $L$ 10 11 12 13 14 15 16 17 $k_0$ 8 9 10 11 12 13 14 15 $\chi_0=\chi^H_{k_0}$ 91.5 91.2 91.13 91.21 91.3 91.49 91.15 91.4
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