April  2020, 40(4): 2105-2134. doi: 10.3934/dcds.2020108

Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system

1. 

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, China 611130

2. 

Department of Mathematics, University of Oklahoma, Norman, USA 73019

3. 

Department of Mathematics, University of South Carolina, Columbia, USA 29208

Corresponding author

Received  March 2019 Revised  November 2019 Published  January 2020

Fund Project: QW acknowledges the support the Fundamental Research Funds for the Central Universities (No. JBK2002002 and No. JBK1805001) and National Natural Science Foundation of China (Grant No. 11501460).

This paper investigates a reaction-advection-diffusion system that describes the evolution of population distributions of two competing species in an enclosed bounded habitat. Here the competition relationships are assumed to be of the Beddington–DeAngelis type. In particular, we consider a situation where first species disperses by a combination of random walk and directed movement along with the population distribution of the second species which disperse randomly within the habitat. We obtain a set of results regarding the qualitative properties of this advective competition system. First of all, we show that this system is globally well-posed and its solutions are classical and uniformly bounded in time. Then we study its steady states in a one-dimensional interval by examining the combined effects of diffusion and advection on the existence and stability of nonconstant positive steady states of the strongly coupled elliptic system. Our stability result of these nontrivial steady states provides a selection mechanism for stable wavemodes of the time-dependent system. Moreover, in the limit of diffusion rates, the steady states of this fully elliptic system can be approximated by nonconstant positive solutions of a shadow system that admits boundary spikes and layers. Furthermore, for the fully elliptic system, we construct solutions with a single boundary spike or an inverted boundary spike, i.e., the first species concentrates on a boundary point while the second species dominates the remaining habitat. These spatial structures model the spatial segregation phenomenon through interspecific competitions. Finally, we perform some numerical simulations to illustrate and support our theoretical findings.

Citation: Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108
References:
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N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.  Google Scholar

[2]

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

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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

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M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

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J. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

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H. Berestycki and P.-L Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

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E. Beretta and Y. Takeuchi, Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delay, SIAM J. Appl. Math., 48 (1988), 627-651.  doi: 10.1137/0148035.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-Deangelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.  Google Scholar

[9]

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.  Google Scholar

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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

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C. CosnerD. DeAngelisJ. S. Ault and D. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65-75.  doi: 10.1006/tpbi.1999.1414.  Google Scholar

[12]

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

[13]

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

[14]

D. DeAngelisR. Goldstein and R. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[15]

M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.  doi: 10.1016/j.jmaa.2004.02.038.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[17]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049.  Google Scholar

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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

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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

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T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.  doi: 10.1016/S0022-247X(02)00395-5.  Google Scholar

[21]

D. IronM. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar

[22]

H. Jin and Z. Wang, Global stability and spatio-temporal patterns of predator-prey systems with density-dependent motion, European J. Appl. Math.. Google Scholar

[23]

Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221.  doi: 10.32917/hmj/1206128382.  Google Scholar

[24]

T. Kato, Study of partial differential equations by means of functional analysis, Springer Classics in Mathematics, (1996). Google Scholar

[25]

J. P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math., 59 (1978), 1-23.  doi: 10.1002/sapm19785911.  Google Scholar

[26]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534-550.  doi: 10.1016/j.jde.2006.08.001.  Google Scholar

[27]

T. KolokolnikovM. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.  Google Scholar

[28]

T. KolokolnikovM. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71.  doi: 10.1111/j.1467-9590.2005.01554.  Google Scholar

[29]

T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion, SIAM J. Appl. Math., 71 (2011), 1428-1457.  doi: 10.1137/100808381.  Google Scholar

[30]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[31]

Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559.  Google Scholar

[32]

M. MaC. Ou and Z. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth, SIAM J. Appl. Math., 72 (2012), 740-766.  doi: 10.1137/110843964.  Google Scholar

[33]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.  doi: 10.2977/prims/1195182020.  Google Scholar

[34]

M. Mimura, Stationary patterns of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  doi: 10.32917/hmj/1206133994.  Google Scholar

[35]

M. MimuraS.-I. Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237.  doi: 10.1007/BF00160536.  Google Scholar

[36]

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

[37]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[38]

W.-M. NiY. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross-diffusion, Discret Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.  Google Scholar

[39]

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

[40]

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

[41]

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.  Google Scholar

[42]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr, 41. American Mathematical Society, Providence, RI, 1995.  Google Scholar

[43]

I. Takagi and W.-M. Ni, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365.  doi: 10.1007/BF03167294.  Google Scholar

[44]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249.  doi: 10.1016/0022-0396(86)90119-1.  Google Scholar

[45]

Y. Takeuchi, Global stability in generalized Lotka-Volterra diffusion systems, J. Math. Anal. Appl., 116 (1986), 209-221.  doi: 10.1016/0022-247X(86)90053-3.  Google Scholar

[46]

B. de Villemereuil and A. Lopez-Sepulcre, Consumer functional responses under intra- and interspecific interference competition, Ecol. Model., 222 (2011), 419-426.  doi: 10.1016/j.ecolmodel.2010.10.011.  Google Scholar

[47]

K. WangQ. Wang and F. Yu, Stationary and time periodic patterns of two-predator and one-prey systems with prey-taxis, Discrete Contin. Dyn. Syst., 37 (2017), 505-543.  doi: 10.3934/dcds.2017021.  Google Scholar

[48]

Q. WangC. 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.  Google Scholar

[49]

Q. WangY. Song and L. 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

[50]

Q. Wang, J. Yang and F. Yu, Global well-posedness of advective Lotka-Volterra competition systems with nonlinear diffusion, Proc. Roy. Soc. Edinburgh Sect. A, (2019). doi: 10.1017/prm.2019.10.  Google Scholar

[51]

Q. Wang and L. Zhang, On the multi-dimensional advective Lotka-Volterra competition systems, Nonlinear Anal. Real World Appl., 37 (2017), 329-349.  doi: 10.1016/j.nonrwa.2017.02.011.  Google Scholar

[52]

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.  Google Scholar

[53]

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.  Google Scholar

[54]

M. Winter and J. Wei, Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity, Differential Integral Equations, 16 (2003), 1153-1180.   Google Scholar

[55]

F. YiJ. Wei and J. 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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.  Google Scholar

[5]

J. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[6]

H. Berestycki and P.-L Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[7]

E. Beretta and Y. Takeuchi, Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delay, SIAM J. Appl. Math., 48 (1988), 627-651.  doi: 10.1137/0148035.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-Deangelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.  Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

C. CosnerD. DeAngelisJ. S. Ault and D. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65-75.  doi: 10.1006/tpbi.1999.1414.  Google Scholar

[12]

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

[13]

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

[14]

D. DeAngelisR. Goldstein and R. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[15]

M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.  doi: 10.1016/j.jmaa.2004.02.038.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[17]

M. A. Herrero and J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.  doi: 10.1007/s002850050049.  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 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

[20]

T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.  doi: 10.1016/S0022-247X(02)00395-5.  Google Scholar

[21]

D. IronM. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar

[22]

H. Jin and Z. Wang, Global stability and spatio-temporal patterns of predator-prey systems with density-dependent motion, European J. Appl. Math.. Google Scholar

[23]

Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221.  doi: 10.32917/hmj/1206128382.  Google Scholar

[24]

T. Kato, Study of partial differential equations by means of functional analysis, Springer Classics in Mathematics, (1996). Google Scholar

[25]

J. P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math., 59 (1978), 1-23.  doi: 10.1002/sapm19785911.  Google Scholar

[26]

W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534-550.  doi: 10.1016/j.jde.2006.08.001.  Google Scholar

[27]

T. KolokolnikovM. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.  Google Scholar

[28]

T. KolokolnikovM. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71.  doi: 10.1111/j.1467-9590.2005.01554.  Google Scholar

[29]

T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion, SIAM J. Appl. Math., 71 (2011), 1428-1457.  doi: 10.1137/100808381.  Google Scholar

[30]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[31]

Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559.  Google Scholar

[32]

M. MaC. Ou and Z. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth, SIAM J. Appl. Math., 72 (2012), 740-766.  doi: 10.1137/110843964.  Google Scholar

[33]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.  doi: 10.2977/prims/1195182020.  Google Scholar

[34]

M. Mimura, Stationary patterns of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  doi: 10.32917/hmj/1206133994.  Google Scholar

[35]

M. MimuraS.-I. Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237.  doi: 10.1007/BF00160536.  Google Scholar

[36]

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

[37]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theor. Biol., 42 (1973), 63-105.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[38]

W.-M. NiY. Wu and Q. Xu, The existence and stability of nontrivial steady states for SKT competition model with cross-diffusion, Discret Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.  Google Scholar

[39]

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

[40]

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

[41]

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.  Google Scholar

[42]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr, 41. American Mathematical Society, Providence, RI, 1995.  Google Scholar

[43]

I. Takagi and W.-M. Ni, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365.  doi: 10.1007/BF03167294.  Google Scholar

[44]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249.  doi: 10.1016/0022-0396(86)90119-1.  Google Scholar

[45]

Y. Takeuchi, Global stability in generalized Lotka-Volterra diffusion systems, J. Math. Anal. Appl., 116 (1986), 209-221.  doi: 10.1016/0022-247X(86)90053-3.  Google Scholar

[46]

B. de Villemereuil and A. Lopez-Sepulcre, Consumer functional responses under intra- and interspecific interference competition, Ecol. Model., 222 (2011), 419-426.  doi: 10.1016/j.ecolmodel.2010.10.011.  Google Scholar

[47]

K. WangQ. Wang and F. Yu, Stationary and time periodic patterns of two-predator and one-prey systems with prey-taxis, Discrete Contin. Dyn. Syst., 37 (2017), 505-543.  doi: 10.3934/dcds.2017021.  Google Scholar

[48]

Q. WangC. 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.  Google Scholar

[49]

Q. WangY. Song and L. 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

[50]

Q. Wang, J. Yang and F. Yu, Global well-posedness of advective Lotka-Volterra competition systems with nonlinear diffusion, Proc. Roy. Soc. Edinburgh Sect. A, (2019). doi: 10.1017/prm.2019.10.  Google Scholar

[51]

Q. Wang and L. Zhang, On the multi-dimensional advective Lotka-Volterra competition systems, Nonlinear Anal. Real World Appl., 37 (2017), 329-349.  doi: 10.1016/j.nonrwa.2017.02.011.  Google Scholar

[52]

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.  Google Scholar

[53]

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.  Google Scholar

[54]

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Figure 1.  The sensitivity function $ \phi(v)\equiv 1 $ is selected. Stable wave mode in the form of $ \cos \frac{k_0\pi x}{L} $, where $ k_0 $ is given in Table 1. $ \chi $ is chosen to be slightly larger than $ \chi_0 $ and the rest system parameters are chosen to be the same as in Table 1. Initial data are small perturbations of $ (\bar u, \bar v) $
Figure 3.  Formation of stable single interior spike of $ u $ and boundary layer of $ v $. Diffusion and advection rates are chosen to be $ D_1 = 5 $, $ \chi = 30 $, $ D_2 = 5\times 10^{-3} $. The rest system parameters are $ a_1 = 0.2 $, $ b_1 = 0.8 $, $ c_1 = 0.1 $ and $ a_2 = 0.6 $, $ b_2 = 0.2 $, $ c_2 = 0.4 $. Initial data are $ u_0 = \bar u+0.5\cos\frac{2\pi x}{5} $ and $ v_0 = \bar v+0.5\cos\frac{2\pi x}{5} $, where $ (\bar u, \bar v) = (0.933...,0.533...) $
Figure 2.  Formation of stable single boundary spike of $ u $ and boundary layer of $ v $. Diffusion and advection rates are chosen to be $ D_1 = 5 $, $ \chi = 30 $, $ D_2 = 5\times 10^{-3} $. The rest system parameters are $ a_1 = 0.2 $, $ b_1 = 0.8 $, $ c_1 = 0.1 $ and $ a_2 = 0.6 $, $ b_2 = 0.2 $, $ c_2 = 0.4 $. Initial data are $ u_0 = \bar u+0.5\cos\frac{2\pi x}{5} $ and $ v_0 = \bar v+0.5\cos\frac{2\pi x}{5} $, where $ (\bar u, \bar v) = (0.933...,0.533...) $
Figure 4.  Formation of stable multiple interior spike of $ u $ and boundary layer of $ v $. Diffusion and advection rates are chosen to be $ D_1 = 5 $, $ \chi = 30 $, $ D_2 = 5\times 10^{-3} $. The rest system parameters are $ a_1 = 0.2 $, $ b_1 = 0.8 $, $ c_1 = 0.1 $ and $ a_2 = 0.6 $, $ b_2 = 0.2 $, $ c_2 = 0.4 $. Initial data are $ u_0 = \bar u+0.5\cos\frac{2\pi x}{5} $ and $ v_0 = \bar v+0.5\cos\frac{2\pi x}{5} $, where $ (\bar u, \bar v) = (0.933...,0.533...) $
Table 1.  Stable wavemode number $ k_0 $ and the corresponding bifurcation value $ \chi_{k_0} $ for different interval length. The system parameters are $ D_1 = 1, D_2 = 0.1, a_1 = a_2 = 0.5, b_1 = 2, b_2 = 1 $ and $ c_1 = 0.5, c_2 = 1 $. According to Proposition 2, the stable wavemode is $ \cos \frac{k_0\pi x}{L} $. Therefore, stable and nontrivial patterns must develop in the form of $ \cos \frac{k_0\pi x}{L} $ if $ \chi $ is chosen to be slightly larger than $ \chi_{k_0} $. We can also see that larger domains support higher wave modes. Figure 1 is given to illustrate the wavemode selection mechanism
Domain size $ L $ 3 5 7 9 11
$ k_0 $ 1 2 2 3 3
$ \chi_k $ 9.9418 10.392 9.9120 9.9418 9.9647
Domain size $ L $ 13 15 17 19 21
$ k_0 $ 4 5 5 6 6
$ \chi_k $ 9.8872 9.9418 9.8937 9.8956 9.9120
Domain size $ L $ 3 5 7 9 11
$ k_0 $ 1 2 2 3 3
$ \chi_k $ 9.9418 10.392 9.9120 9.9418 9.9647
Domain size $ L $ 13 15 17 19 21
$ k_0 $ 4 5 5 6 6
$ \chi_k $ 9.8872 9.9418 9.8937 9.8956 9.9120
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