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doi: 10.3934/dcdsb.2021277
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Dynamic transitions for the S-K-T competition system

1. 

School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China

2. 

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

* Corresponding author: Dongpei Zhang

Received  March 2021 Early access November 2021

Fund Project: The work is supported by the Young Scholars Development Fund of SWPU (Grant No. 201899010079), the scientific research starting project of SWPU (Grant No. 2018QHZ029), Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and Its Applications (No. 18TD0013), Southwest Petroleum Univ. under Grant 2019CXTD08, Youth Science and Technology Innovation Team of Southwest Petroleum Univ. for Nonlinear Systems (No. 2017CXTD02)

This paper is concerned with dynamical transition for biological competition system modeled by the S-K-T equations. We study the dynamical behaviour of the S-K-T equations with two different boundary conditions. For the system under non-homogeneous Dirichlet boundary condition, we show that the system undergoes a mixed dynamic transition from the homogeneous state to steady state solutions when the bifurcation parameter cross the critical surface. For the system with Neumann boundary condition, we prove that the system undergoes a mixed dynamic transition, a jump transition and a continuous transition when the bifurcation parameter cross the critical number. Finally, two examples are provided to validate the effectiveness of the theoretical results.

Citation: Ruikuan Liu, Dongpei Zhang. Dynamic transitions for the S-K-T competition system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021277
References:
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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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Y. ChoiR. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.  doi: 10.3934/dcds.2004.10.719.  Google Scholar

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D. Deng and R. Liu, Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3175-3193.  doi: 10.3934/dcdsb.2018306.  Google Scholar

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C.-H. HsiaC.-S. LinT. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A., 471 (2015), 1-24.  doi: 10.1098/rspa.2014.0353.  Google Scholar

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K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858.  doi: 10.1016/j.jde.2014.11.016.  Google Scholar

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R. Liu and Q. Wang, $S^1$ attractor bifurcation analysis for an electrically conducting fluid flows between two concentric axial cylinders, Phys. D, 392 (2019), 17-33.  doi: 10.1016/j.physd.2019.03.001.  Google Scholar

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Y. LouW. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.  Google Scholar

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C. LuY. MaoQ. Wang and D. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differential Equations, 267 (2019), 2560-2593.  doi: 10.1016/j.jde.2019.03.021.  Google Scholar

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[21]

T. Ma and S. Wang, Dynamic phase transitions for ferromagnetic systems, J. Math. Phys., 49 (2008), 1-18.  doi: 10.1063/1.2913504.  Google Scholar

[22]

T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[23]

T. Ma and S. Wang, Phase transitions for Belousov-Zhabotinsky reactions, Math. Methods Appl. Sci., 34 (2011), 1381-1397.  doi: 10.1002/mma.1446.  Google Scholar

[24]

T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 1-23.  doi: 10.1063/1.3559120.  Google Scholar

[25]

T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[26]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[27]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

L. SunJ. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9.  Google Scholar

[32]

Q.-J. Tan, A free boundary problem describing S-K-T competition ecological model with cross-diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 53-82.  doi: 10.1016/j.nonrwa.2018.06.010.  Google Scholar

[33]

L. WangY. Wu and Q. Xu, Instability of spiky steady states for S-K-T biological competing model with cross-diffusion, Nonlinear Anal., 159 (2017), 424-457.  doi: 10.1016/j.na.2017.02.026.  Google Scholar

[34]

Q. Wang, On the steady states of a shadow system of the SKT competition system, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2941-2961.  doi: 10.3934/dcdsb.2014.19.2941.  Google Scholar

[35]

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

[36]

Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.  doi: 10.1016/j.jde.2004.08.015.  Google Scholar

[37]

Y. Wu and Q. Xu, The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.  doi: 10.3934/dcds.2011.29.367.  Google Scholar

[38]

Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross-diffuion systems, Phys. D, 200 (2005), 325-358.  doi: 10.1016/j.physd.2004.11.010.  Google Scholar

[39]

A. Yagi, Exponential attractors for competing species model with cross-diffusions, Discrete Contin. Dyn. Syst., 22 (2008), 1091-1120.  doi: 10.3934/dcds.2008.22.1091.  Google Scholar

[40]

Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 282–299. doi: 10.1142/9789812834744_0013.  Google Scholar

[41]

D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Math. Method Appl. Sci., 41 (2018), 4641-4658.  doi: 10.1002/mma.4919.  Google Scholar

[42]

X. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differential Equations, 55 (2016), 1-25.  doi: 10.1007/s00526-016-1021-8.  Google Scholar

show all references

References:
[1]

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

[2]

Y. ChoiR. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.  doi: 10.3934/dcds.2004.10.719.  Google Scholar

[3]

S. Choo and Y.-H. Kim, Global stability in a discrete Lotka-Volterra competition model, J. Comput. Anal. Appl., 23 (2017), 276-293.   Google Scholar

[4]

D. Deng and R. Liu, Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3175-3193.  doi: 10.3934/dcdsb.2018306.  Google Scholar

[5]

P. Du and R. Temam, Weak solutions of the Shigesada-Kawasaki-Teramoto equations and their attractors, Nonlinear Anal., 159 (2017), 339-364.  doi: 10.1016/j.na.2017.01.017.  Google Scholar

[6]

W. Feng, Competitive exclusion and persistence in models of resource and sexual competition, J. Math. Biol., 35 (1997), 683-694.  doi: 10.1007/s002850050071.  Google Scholar

[7]

C.-H. HsiaC.-S. LinT. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A., 471 (2015), 1-24.  doi: 10.1098/rspa.2014.0353.  Google Scholar

[8]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.   Google Scholar

[9]

K. Kishimota and H.-F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion system on convex domain, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[10]

K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858.  doi: 10.1016/j.jde.2014.11.016.  Google Scholar

[11]

A. Leung, Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73 (1980), 204-218.  doi: 10.1016/0022-247X(80)90028-1.  Google Scholar

[12]

A. Leung and D. Clark, Bifurcations and large-time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data, J. Differential Equations, 35 (1980), 113-127.  doi: 10.1016/0022-0396(80)90052-2.  Google Scholar

[13]

L. Li and R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations, 4 (1991), 817-834.   Google Scholar

[14]

R. Liu and Q. Wang, $S^1$ attractor bifurcation analysis for an electrically conducting fluid flows between two concentric axial cylinders, Phys. D, 392 (2019), 17-33.  doi: 10.1016/j.physd.2019.03.001.  Google Scholar

[15]

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

[16]

Y. LouW. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dynam. Systems, 4 (1998), 193-203.  doi: 10.3934/dcds.1998.4.193.  Google Scholar

[17]

Y. LouW. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.  Google Scholar

[18]

C. LuY. MaoQ. Wang and D. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differential Equations, 267 (2019), 2560-2593.  doi: 10.1016/j.jde.2019.03.021.  Google Scholar

[19]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science Series A: Volume 53, World Scientific Publishing Co. Pte. Ltd., 2005. doi: 10.1142/9789812701152.  Google Scholar

[20]

T. Ma and S. Wang, Stability and bifurcation of the Taylor problem, Arch. Ration. Mech. Anal., 181 (2006), 149-176.  doi: 10.1007/s00205-006-0415-8.  Google Scholar

[21]

T. Ma and S. Wang, Dynamic phase transitions for ferromagnetic systems, J. Math. Phys., 49 (2008), 1-18.  doi: 10.1063/1.2913504.  Google Scholar

[22]

T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[23]

T. Ma and S. Wang, Phase transitions for Belousov-Zhabotinsky reactions, Math. Methods Appl. Sci., 34 (2011), 1381-1397.  doi: 10.1002/mma.1446.  Google Scholar

[24]

T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 1-23.  doi: 10.1063/1.3559120.  Google Scholar

[25]

T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[26]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[27]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

L. SunJ. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9.  Google Scholar

[32]

Q.-J. Tan, A free boundary problem describing S-K-T competition ecological model with cross-diffusion, Nonlinear Anal. Real World Appl., 45 (2019), 53-82.  doi: 10.1016/j.nonrwa.2018.06.010.  Google Scholar

[33]

L. WangY. Wu and Q. Xu, Instability of spiky steady states for S-K-T biological competing model with cross-diffusion, Nonlinear Anal., 159 (2017), 424-457.  doi: 10.1016/j.na.2017.02.026.  Google Scholar

[34]

Q. Wang, On the steady states of a shadow system of the SKT competition system, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2941-2961.  doi: 10.3934/dcdsb.2014.19.2941.  Google Scholar

[35]

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

[36]

Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340.  doi: 10.1016/j.jde.2004.08.015.  Google Scholar

[37]

Y. Wu and Q. Xu, The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.  doi: 10.3934/dcds.2011.29.367.  Google Scholar

[38]

Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross-diffuion systems, Phys. D, 200 (2005), 325-358.  doi: 10.1016/j.physd.2004.11.010.  Google Scholar

[39]

A. Yagi, Exponential attractors for competing species model with cross-diffusions, Discrete Contin. Dyn. Syst., 22 (2008), 1091-1120.  doi: 10.3934/dcds.2008.22.1091.  Google Scholar

[40]

Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 282–299. doi: 10.1142/9789812834744_0013.  Google Scholar

[41]

D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Math. Method Appl. Sci., 41 (2018), 4641-4658.  doi: 10.1002/mma.4919.  Google Scholar

[42]

X. Zhao and P. Zhou, On a Lotka-Volterra competition model: The effects of advection and spatial variation, Calc. Var. Partial Differential Equations, 55 (2016), 1-25.  doi: 10.1007/s00526-016-1021-8.  Google Scholar

Figure 1.  cross-section graph of $ \Lambda_k $
Figure 2.  The dynamical transitions of $ (13) $ when $ \int_{\Omega}\psi_{1}^{3}dx \neq 0 $
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