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

Ke Wang; Qi Wang; Feng Yu

doi:10.3934/dcds.2017021

Previous Article
Random attractor for stochastic non-autonomous damped wave equation with critical exponent
DCDS Home
This Issue
Next Article
Monotone dynamical systems: Reflections on new advances & applications

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

Ke Wang , Qi Wang ^, and Feng Yu ^†,

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.

Keywords: Pattern formation, predator-prey model, prey-taxis, stationary solutions, time-periodic solutions, stability analysis.

Mathematics Subject Classification: Primary:35B36, 92D25, 35B10, 35B32;Secondary:35B35, 35J47, 35K20, 35Q9.

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:

[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. 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.
[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. 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.
[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.
[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. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[55]	T. Tona and N. Hieu, Dynamics of species in a model with two predators and one prey, Nonlinear Anal., 74 (2011), 4868-4881. doi: 10.1016/j.na.2011.04.061.
[56]	P. Turchin, Quantitative Analysis of Movement, Sinauer, Sunderland, Mass. , 1998.
[57]	Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.
[58]	Q. Wang, Y. Song and L. Shao, Nonconstant positive steady states and pattern formation of 1D prey-taxis systems, J. Nonlinear Sci., (2016), 1-27. doi: 10.1007/s00332-016-9326-5.
[59]	Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species keller-segel chemotaxis model: effect of cellular growth, preprint, http://arxiv.org/abs/1505.06463.
[60]	Q. Wang, L. Zhang, J. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807. doi: 10.3934/krm.2015.8.777.
[61]	X. Wang, W. Wang and G. 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.
[62]	X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.
[63]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[64]	D. Xiao and S. Ruan, Codimension two bifurcations in a predator-prey system with group defense, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 2123-2131. doi: 10.1142/S021812740100336X.
[65]	J. Zhou, C.-G. Kim and J. Shi, Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type Ⅱ functional response and cross-diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 3875-3899. doi: 10.3934/dcds.2014.34.3875.

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. 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.
[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. 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.
[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.
[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. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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. 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.
[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. 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.
[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. 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.
[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.
[55]	T. Tona and N. Hieu, Dynamics of species in a model with two predators and one prey, Nonlinear Anal., 74 (2011), 4868-4881. doi: 10.1016/j.na.2011.04.061.
[56]	P. Turchin, Quantitative Analysis of Movement, Sinauer, Sunderland, Mass. , 1998.
[57]	Q. Wang, C. Gai and J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284. doi: 10.3934/dcds.2015.35.1239.
[58]	Q. Wang, Y. Song and L. Shao, Nonconstant positive steady states and pattern formation of 1D prey-taxis systems, J. Nonlinear Sci., (2016), 1-27. doi: 10.1007/s00332-016-9326-5.
[59]	Q. Wang, J. Yang and L. Zhang, Time periodic and stable patterns of a two-competing-species keller-segel chemotaxis model: effect of cellular growth, preprint, http://arxiv.org/abs/1505.06463.
[60]	Q. Wang, L. Zhang, J. Yang and J. Hu, Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807. doi: 10.3934/krm.2015.8.777.
[61]	X. Wang, W. Wang and G. 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.
[62]	X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.
[63]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[64]	D. Xiao and S. Ruan, Codimension two bifurcations in a predator-prey system with group defense, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 2123-2131. doi: 10.1142/S021812740100336X.
[65]	J. Zhou, C.-G. Kim and J. Shi, Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type Ⅱ functional response and cross-diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 3875-3899. doi: 10.3934/dcds.2014.34.3875.

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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

$k$	1	2	3	4	5	6	7	8	9
$\chi^S_{k}$	66.98	18.98	10.30	7.49	6.41	6.05	6.09	6.36	6.80
$\chi^H_{k}$	1204.20	550.84	504.20	575.80	705.50	878.48	1089.70	1336.90	1619.19

Table Options

Download as excel

$k$	1	2	3	4	5	6	7	8	9
$\chi^S_{k}$	66.98	18.98	10.30	7.49	6.41	6.05	6.09	6.36	6.80
$\chi^H_{k}$	1204.20	550.84	504.20	575.80	705.50	878.48	1089.70	1336.90	1619.19

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 $L$	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

Table Options

Download as excel

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

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

$k$	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	236.80	278.76
$\chi^H_{k}$	186.37	96.73	92.57	105.46	127.03	155.24	189.40	229.24	274.64

Table Options

Download as excel

$k$	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	236.80	278.76
$\chi^H_{k}$	186.37	96.73	92.57	105.46	127.03	155.24	189.40	229.24	274.64

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 $L$	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	91.30	91.49	91.15	91.40

Table Options

Download as excel

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	91.30	91.49	91.15	91.40

[1]	Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162
[2]	Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045
[3]	Xin Wang, Ruijing Li, Yu Shi. Global generalized solutions to a three species predator-prey model with prey-taxis. Discrete and Continuous Dynamical Systems - B, 2022, 27 (12) : 7021-7042. doi: 10.3934/dcdsb.2022031
[4]	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
[5]	Guoqiang Ren, Bin Liu. Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 759-779. doi: 10.3934/dcds.2021136
[6]	Mostafa Bendahmane. Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis. Networks and Heterogeneous Media, 2008, 3 (4) : 863-879. doi: 10.3934/nhm.2008.3.863
[7]	Dan Li. Global stability in a multi-dimensional predator-prey system with prey-taxis. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1681-1705. doi: 10.3934/dcds.2020337
[8]	Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure and Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133
[9]	Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823
[10]	Xiaoying Wang, Xingfu Zou. Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Mathematical Biosciences & Engineering, 2018, 15 (3) : 775-805. doi: 10.3934/mbe.2018035
[11]	Henri Berestycki, Alessandro Zilio. Predator-prey models with competition, Part Ⅲ: Classification of stationary solutions. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7141-7162. doi: 10.3934/dcds.2019299
[12]	Wonlyul Ko, Inkyung Ahn. Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction. Communications on Pure and Applied Analysis, 2018, 17 (2) : 375-389. doi: 10.3934/cpaa.2018021
[13]	Wan-Tong Li, Yong-Hong Fan. Periodic solutions in a delayed predator-prey models with nonmonotonic functional response. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 175-185. doi: 10.3934/dcdsb.2007.8.175
[14]	Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267
[15]	Meng Fan, Qian Wang. Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 563-574. doi: 10.3934/dcdsb.2004.4.563
[16]	Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268
[17]	Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138
[18]	Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265
[19]	Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[20]	Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

2021 Impact Factor: 1.588

Tools

Metrics

Tools

Article outline

Figures and Tables

2021 Impact Factor: 1.588

Tools

Figures and Tables

2021 Impact Factor: 1.588

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors